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# Knight's tour

(Redirected from Knight's Tour)
Knight's tour
You are encouraged to solve this task according to the task description, using any language you may know.

Problem: you have a standard 8x8 chessboard, empty but for a single knight on some square. Your task is to emit a series of legal knight moves that result in the knight visiting every square on the chessboard exactly once. Note that it is not a requirement that the tour be "closed"; that is, the knight need not end within a single move of its start position.

Input and output may be textual or graphical, according to the conventions of the programming environment. If textual, squares should be indicated in algebraic notation. The output should indicate the order in which the knight visits the squares, starting with the initial position. The form of the output may be a diagram of the board with the squares numbered according to visitation sequence, or a textual list of algebraic coordinates in order, or even an actual animation of the knight moving around the chessboard.

Input: starting square

Output: move sequence

## 11l

Translation of: Python
V _kmoves = [(2, 1), (1, 2), (-1, 2), (-2, 1), (-2, -1), (-1, -2), (1, -2), (2, -1)]

F chess2index(=chess, boardsize)
‘Convert Algebraic chess notation to internal index format’
chess = chess.lowercase()
V x = chess[0].code - ‘a’.code
V y = boardsize - Int(chess[1..])
R (x, y)

F boardstring(board, boardsize)
V r = 0 .< boardsize
V lines = ‘’
L(y) r
lines ‘’= "\n"r.map(x -> (I @board[(x, @y)] {‘#2’.format(@board[(x, @y)])} E ‘ ’)).join(‘,’)
R lines

F knightmoves(board, P, boardsize)
V (Px, Py) = P
V kmoves = Set(:_kmoves.map((x, y) -> (@Px + x, @Py + y)))
kmoves = Set(Array(kmoves).filter((x, y) -> x C 0 .< @boardsize & y C 0 .< @boardsize & [email protected][(x, y)]))
R kmoves

F accessibility(board, P, boardsize)
[(Int, (Int, Int))] access
V brd = copy(board)
L(pos) knightmoves(board, P, boardsize' boardsize)
brd[pos] = -1
access.append((knightmoves(brd, pos, boardsize' boardsize).len, pos))
brd[pos] = 0
R access

F knights_tour(start, boardsize, _debug = 0B)
[(Int, Int) = Int] board
L(x) 0 .< boardsize
L(y) 0 .< boardsize
board[(x, y)] = 0
V move = 1
V P = chess2index(start, boardsize)
board[P] = move
move++
I _debug
print(boardstring(board, boardsize' boardsize))
L move <= board.len
P = min(accessibility(board, P, boardsize))[1]
board[P] = move
move++
I _debug
print(boardstring(board, boardsize' boardsize))
input("\n#2 next: ".format(move))
R board

L(boardsize, start) [(5, ‘c3’), (8, ‘h8’), (10, ‘e6’)]
print(‘boardsize: ’boardsize)
print(‘Start position: ’start)
V board = knights_tour(start, boardsize)
print(boardstring(board, boardsize' boardsize))
print()
Output:
boardsize: 5
Start position: c3

19,12,17, 6,21
2, 7,20,11,16
13,18, 1,22, 5
8, 3,24,15,10
25,14, 9, 4,23

boardsize: 8
Start position: h8

38,41,18, 3,22,27,16, 1
19, 4,39,42,17, 2,23,26
40,37,54,21,52,25,28,15
5,20,43,56,59,30,51,24
36,55,58,53,44,63,14,29
9, 6,45,62,57,60,31,50
46,35, 8,11,48,33,64,13
7,10,47,34,61,12,49,32

boardsize: 10
Start position: e6

29, 4,57,24,73, 6,95,10,75, 8
58,23,28, 5,94,25,74, 7,100,11
3,30,65,56,27,72,99,96, 9,76
22,59, 2,63,68,93,26,81,12,97
31,64,55,66, 1,82,71,98,77,80
54,21,60,69,62,67,92,79,88,13
49,32,53,46,83,70,87,42,91,78
20,35,48,61,52,45,84,89,14,41
33,50,37,18,47,86,39,16,43,90
36,19,34,51,38,17,44,85,40,15

## 360 Assembly

Translation of: BBC PASIC
*        Knight's tour             20/03/2017
KNIGHT CSECT
USING KNIGHT,R13 base registers
B 72(R15) skip savearea
DC 17F'0' savearea
STM R14,R12,12(R13) save previous context
MVC PG(20),=CL20'Knight''s tour ..x..'
L R1,NN n
XDECO R1,XDEC edit
MVC PG+14(2),XDEC+10 n
MVC PG+17(2),XDEC+10 n
XPRNT PG,L'PG print buffer
LA R0,1 1
ST R0,X x=1
ST R0,Y y=1
SR R0,R0 0
ST R0,TOTAL total=0
LOOP EQU * do loop
L R1,X x
BCTR R1,0 -1
MH R1,NNH *n
L R0,Y y
BCTR R0,0 -1
AR R1,R0 (x-1)*n+y-1
SLA R1,1 ((x-1)*n+y-1)*2
LA R0,1 1
STH R0,BOARD(R1) board(x,y)=1
L R2,TOTAL total
LA R2,1(R2) total+1
STH R2,DISP(R1) disp(x,y)=total+1
ST R2,TOTAL total=total+1
L R1,X x
L R2,Y y
BAL R14,CHOOSEMV call choosemv(x,y)
C R0,=F'0' until(choosemv(x,y)=0)
BNE LOOP loop
LA R2,KN*KN n*n
IF C,R2,NE,TOTAL THEN if total<>n*n then
XPRNT =C'error!!',7 print error
ENDIF , endif
LA R6,1 i=1
DO WHILE=(C,R6,LE,NN) do i=1 to n
MVC PG,=CL128' ' init buffer
LA R10,PG pgi=0
LA R7,1 j=1
DO WHILE=(C,R7,LE,NN) do j=1 to n
LR R1,R6 i
BCTR R1,0 -1
MH R1,NNH *n
LR R0,R7 j
BCTR R0,0 -1
AR R1,R0 (i-1)*n+j-1
SLA R1,1 ((i-1)*n+j-1)*2
LH R2,DISP(R1) disp(i,j)
XDECO R2,XDEC edit
MVC 0(4,R10),XDEC+8 output
LA R10,4(R10) pgi+=4
LA R7,1(R7) j++
ENDDO , enddo j
XPRNT PG,L'PG print buffer
LA R6,1(R6) i++
ENDDO , enddo i
L R13,4(0,R13) restore previous savearea pointer
LM R14,R12,12(R13) restore previous context
XR R15,R15 return_code=0
BR R14 exit
*------- ---- ----------------------------------------
CHOOSEMV EQU * choosemv(xc,yc)
ST R14,SAVEACMV save return point
ST R1,XC store xc
ST R2,YC store yc
MVC MM,=F'9' m=9
L R1,XC xc
LA R1,1(R1)
L R2,YC yc
LA R2,2(R2)
BAL R14,TRYMV call trymv(xc+1,yc+2)
L R1,XC xc
LA R1,1(R1)
L R2,YC yc
SH R2,=H'2'
BAL R14,TRYMV call trymv(xc+1,yc-2)
L R1,XC xc
BCTR R1,0
L R2,YC yc
LA R2,2(R2)
BAL R14,TRYMV call trymv(xc-1,yc+2)
L R1,XC xc
BCTR R1,0
L R2,YC yc
SH R2,=H'2'
BAL R14,TRYMV call trymv(xc-1,yc-2)
L R1,XC xc
LA R1,2(R1)
L R2,YC yc
LA R2,1(R2)
BAL R14,TRYMV call trymv(xc+2,yc+1)
L R1,XC xc
LA R1,2(R1)
L R2,YC yc
BCTR R2,0
BAL R14,TRYMV call trymv(xc+2,yc-1)
L R1,XC xc
SH R1,=H'2'
L R2,YC yc
LA R2,1(R2)
BAL R14,TRYMV call trymv(xc-2,yc+1)
L R1,XC xc
SH R1,=H'2'
L R2,YC yc
BCTR R2,0
BAL R14,TRYMV call trymv(xc-2,yc-1)
L R4,MM m
IF C,R4,EQ,=F'9' THEN if m=9 then
LA R0,0 return(0)
ELSE , else
MVC X,NEWX x=newx
MVC Y,NEWY y=newy
LA R0,1 return(1)
ENDIF , endif
L R14,SAVEACMV restore return point
BR R14 return
SAVEACMV DS A return point
*------- ---- ----------------------------------------
TRYMV EQU * trymv(xt,yt)
ST R14,SAVEATMV save return point
ST R1,XT store xt
ST R2,YT store yt
SR R10,R10 n=0
BAL R14,VALIDMV
IF LTR,R0,Z,R0 THEN if validmv(xt,yt)=0 then
LA R0,0 return(0)
B RETURTMV
ENDIF , endif
L R1,XT
LA R1,1(R1) xt+1
L R2,YT
LA R2,2(R2) yt+2
BAL R14,VALIDMV
IF C,R0,EQ,=F'1' THEN if validmv(xt+1,yt+2)=1 then
LA R10,1(R10) n=n+1;
ENDIF , endif
L R1,XT
LA R1,1(R1) xt+1
L R2,YT
SH R2,=H'2' yt-2
BAL R14,VALIDMV
IF C,R0,EQ,=F'1' THEN if validmv(xt+1,yt-2)=1 then
LA R10,1(R10) n=n+1;
ENDIF , endif
L R1,XT
BCTR R1,0 xt-1
L R2,YT
LA R2,2(R2) yt+2
BAL R14,VALIDMV
IF C,R0,EQ,=F'1' THEN if validmv(xt-1,yt+2)=1 then
LA R10,1(R10) n=n+1;
ENDIF , endif
L R1,XT
BCTR R1,0 xt-1
L R2,YT
SH R2,=H'2' yt-2
BAL R14,VALIDMV
IF C,R0,EQ,=F'1' THEN if validmv(xt-1,yt-2)=1 then
LA R10,1(R10) n=n+1;
ENDIF , endif
L R1,XT
LA R1,2(R1) xt+2
L R2,YT
LA R2,1(R2) yt+1
BAL R14,VALIDMV
IF C,R0,EQ,=F'1' THEN if validmv(xt+2,yt+1)=1 then
LA R10,1(R10) n=n+1;
ENDIF , endif
L R1,XT
LA R1,2(R1) xt+2
L R2,YT
BCTR R2,0 yt-1
BAL R14,VALIDMV
IF C,R0,EQ,=F'1' THEN if validmv(xt+2,yt-1)=1 then
LA R10,1(R10) n=n+1;
ENDIF , endif
L R1,XT
SH R1,=H'2' xt-2
L R2,YT
LA R2,1(R2) yt+1
BAL R14,VALIDMV
IF C,R0,EQ,=F'1' THEN if validmv(xt-2,yt+1)=1 then
LA R10,1(R10) n=n+1;
ENDIF , endif
L R1,XT
SH R1,=H'2' xt-2
L R2,YT
BCTR R2,0 yt-1
BAL R14,VALIDMV
IF C,R0,EQ,=F'1' THEN if validmv(xt-2,yt-1)=1 then
LA R10,1(R10) n=n+1;
ENDIF , endif
IF C,R10,LT,MM THEN if n<m then
ST R10,MM m=n
MVC NEWX,XT newx=xt
MVC NEWY,YT newy=yt
ENDIF , endif
RETURTMV L R14,SAVEATMV restore return point
BR R14 return
SAVEATMV DS A return point
*------- ---- ----------------------------------------
VALIDMV EQU * validmv(xv,yv)
C R1,=F'1' if xv<1 then
BL RET0
C R1,NN if xv>nn then
BH RET0
C R2,=F'1' if yv<1 then
BL RET0
C R2,NN if yv>nn then
BNH OK
RET0 SR R0,R0 return(0)
B RETURVMV
OK LR R3,R1 xv
BCTR R3,0
MH R3,NNH *n
LR R0,R2 yv
BCTR R0,0
AR R3,R0
SLA R3,1
LH R4,BOARD(R3) board(xv,yv)
IF LTR,R4,Z,R4 THEN if board(xv,yv)=0 then
LA R0,1 return(1)
ELSE , else
SR R0,R0 return(0)
ENDIF , endif
RETURVMV BR R14 return
* ---- ----------------------------------------
KN EQU 8 n compile-time
NN DC A(KN) n fullword
NNH DC AL2(KN) n halfword
BOARD DC (KN*KN)H'0' dim board(n,n) init 0
DISP DC (KN*KN)H'0' dim disp(n,n) init 0
X DS F
Y DS F
TOTAL DS F
XC DS F
YC DS F
MM DS F
NEWX DS F
NEWY DS F
XT DS F
YT DS F
XDEC DS CL12
PG DC CL128' ' buffer
YREGS
END KNIGHT
Output:
Knight's tour  8x 8
1   4  57  20  47   6  49  22
34  19   2   5  58  21  46   7
3  56  35  60  37  48  23  50
18  33  38  55  52  59   8  45
39  14  53  36  61  44  51  24
32  17  40  43  54  27  62   9
13  42  15  30  11  64  25  28
16  31  12  41  26  29  10  63

First, we specify a naive implementation the package Knights_Tour with naive backtracking. It is a bit more general than required for this task, by providing a mechanism not to visit certain coordinates. This mechanism is actually useful for the task Solve a Holy Knight's tour#Ada, which also uses the package Knights_Tour.

generic
Size: Integer;
package Knights_Tour is

subtype Index is Integer range 1 .. Size;
type Tour is array (Index, Index) of Natural;
Empty: Tour := (others => (others => 0));

-- finds tour via backtracking
-- either no tour has been found, i.e., Get_Tour returns Scene
-- or the Result(X,Y)=K if and only if I,J is visited at the K-th move
-- for all X, Y, Scene(X,Y) must be either 0 or Natural'Last,
-- where Scene(X,Y)=Natural'Last means "don't visit coordiates (X,Y)!"

function Count_Moves(Board: Tour) return Natural;
-- counts the number of possible moves, i.e., the number of 0's on the board

procedure Tour_IO(The_Tour: Tour; Width: Natural := 4);
-- writes The_Tour to the output using Ada.Text_IO;

end Knights_Tour;

Here is the implementation:

package body Knights_Tour is

type Pair is array(1..2) of Integer;
type Pair_Array is array (Positive range <>) of Pair;

Pairs: constant Pair_Array (1..8)
:= ((-2,1),(-1,2),(1,2),(2,1),(2,-1),(1,-2),(-1,-2),(-2,-1));
-- places for the night to go (relative to the current position)

function Count_Moves(Board: Tour) return Natural is
N: Natural := 0;
begin
for I in Index loop
for J in Index loop
if Board(I,J) < Natural'Last then
N := N + 1;
end if;
end loop;
end loop;
return N;
end Count_Moves;

function Get_Tour(Start_X, Start_Y: Index; Scene: Tour := Empty)
Done: Boolean;
Move_Count: Natural := Count_Moves(Scene);
Visited: Tour;

-- Visited(I, J) = 0: not yet visited
-- Visited(I, J) = K: visited at the k-th move
-- Visited(I, J) = Integer'Last: never visit

procedure Visit(X, Y: Index; Move_Number: Positive; Found: out Boolean) is
XX, YY: Integer;
begin
Found := False;
Visited(X, Y) := Move_Number;
if Move_Number = Move_Count then
Found := True;
else
for P in Pairs'Range loop
XX := X + Pairs(P)(1);
YY := Y + Pairs(P)(2);
if (XX in Index) and then (YY in Index)
and then Visited(XX, YY) = 0 then
Visit(XX, YY, Move_Number+1, Found); -- recursion
if Found then
return; -- no need to search further
end if;
end if;
end loop;
Visited(X, Y) := 0; -- undo previous mark
end if;
end Visit;

begin
Visited := Scene;
Visit(Start_X, Start_Y, 1, Done);
if not Done then
Visited := Scene;
end if;
return Visited;
end Get_Tour;

procedure Tour_IO(The_Tour: Tour; Width: Natural := 4) is
begin
for I in Index loop
for J in Index loop
if The_Tour(I, J) < Integer'Last then
else
for W in 1 .. Width-1 loop
end loop;
end if;
end loop;
end loop;
end Tour_IO;

end Knights_Tour;

Here is the main program:

procedure Test_Knight is

package KT is new Knights_Tour(Size => Size);

begin
KT.Tour_IO(KT.Get_Tour(1, 1));
end Test_Knight;

For small sizes, this already works well (< 1 sec for size 8). Sample output:

>./test_knight 8
1  38  55  34   3  36  19  22
54  47   2  37  20  23   4  17
39  56  33  46  35  18  21  10
48  53  40  57  24  11  16   5
59  32  45  52  41  26   9  12
44  49  58  25  62  15   6  27
31  60  51  42  29   8  13  64
50  43  30  61  14  63  28   7

For larger sizes we'll use Warnsdorff's heuristic (without any thoughtful tie breaking). We enhance the specification adding a function Warnsdorff_Get_Tour. This enhancement of the package Knights_Tour will also be used for the task Solve a Holy Knight's tour#Ada. The specification of Warnsdorff_Get_Tour is the following.

function Warnsdorff_Get_Tour(Start_X, Start_Y: Index; Scene: Tour := Empty)
-- uses Warnsdorff heurisitic to find a tour faster
-- same interface as Get_Tour

Its implementation is as follows.

function Warnsdorff_Get_Tour(Start_X, Start_Y: Index;  Scene: Tour := Empty)
Done: Boolean;
Visited: Tour; -- see comments from Get_Tour above
Move_Count: Natural := Count_Moves(Scene);

function Neighbors(X, Y: Index) return Natural is
Result: Natural := 0;
begin
for P in Pairs'Range loop
if X+Pairs(P)(1) in Index and then Y+Pairs(P)(2) in Index and then
Visited(X+Pairs(P)(1), Y+Pairs(P)(2)) = 0 then
Result := Result + 1;
end if;
end loop;
return Result;
end Neighbors;

procedure Sort(Options: in out Pair_Array) is
N_Bors: array(Options'Range) of Natural;
K: Positive range Options'Range;
N: Natural;
P: Pair;
begin
for Opt in Options'Range loop
N_Bors(Opt) := Neighbors(Options(Opt)(1), Options(Opt)(2));
end loop;
for Opt in Options'Range loop
K := Opt;
for Alternative in Opt+1 .. Options'Last loop
if N_Bors(Alternative) < N_Bors(Opt) then
K := Alternative;
end if;
end loop;
N  := N_Bors(Opt);
N_Bors(Opt) := N_Bors(K);
N_Bors(K)  := N;
P  := Options(Opt);
Options(Opt) := Options(K);
Options(K)  := P;
end loop;
end Sort;

procedure Visit(X, Y: Index; Move: Positive; Found: out Boolean) is
Next_Count: Natural range 0 .. 8 := 0;
Next_Steps: Pair_Array(1 .. 8);
XX, YY: Integer;
begin
Found := False;
Visited(X, Y) := Move;
if Move = Move_Count then
Found := True;
else
-- consider all possible places to go
for P in Pairs'Range loop
XX := X + Pairs(P)(1);
YY := Y + Pairs(P)(2);
if (XX in Index) and then (YY in Index)
and then Visited(XX, YY) = 0 then
Next_Count := Next_Count+1;
Next_Steps(Next_Count) := (XX, YY);
end if;
end loop;

Sort(Next_Steps(1 .. Next_Count));

for N in 1 .. Next_Count loop
Visit(Next_Steps(N)(1), Next_Steps(N)(2), Move+1, Found);
if Found then
return; -- no need to search further
end if;
end loop;

-- if we didn't return above, we have to undo our move
Visited(X, Y) := 0;
end if;
end Visit;

begin
Visited := Scene;
Visit(Start_X, Start_Y, 1, Done);
if not Done then
Visited := Scene;
end if;
return Visited;
end Warnsdorff_Get_Tour;

The modification for the main program is trivial:

procedure Test_Fast is

package KT is new Knights_Tour(Size => Size);

begin
KT.Tour_IO(KT.Warnsdorff_Get_Tour(1, 1));
end Test_Fast;

This works still well for somewhat larger sizes:

>./test_fast 24
1 108  45  52   3 112  57  60   5  62 131 144   7  64 147 170   9  66 187 192  11  68  71 190
46  51   2 111  56  53   4 113 130  59   6  63 146 169   8  65 186 215  10  67 188 191  12  69
107  44 109  54 123 114 129  58  61 132 145 168 143 148 185 214 171 198 225 216 193  70 189  72
50  47 122 115 110  55 140 133 128 167 142 149 184 213 172 199 226 255 246 197 224 217 194  13
43 106  49 124 139 134 127 166 141 150 183 212 173 200 227 254 247 242 223 256 245 196  73 218
48 121 116 135 126 165 138 151 182 211 174 201 228 253 248 241 290 263 304 243 222 257  14 195
105  42 125 164 137 152 181 210 175 202 229 252 249 240 289 264 329 308 291 262 303 244 219  74
120 117 136 153 180 163 176 203 230 267 250 239 288 265 328 309 334 345 330 305 292 221 258  15
41 104 119 160 177 204 231 268 209 238 287 266 251 310 335 344 357 332 307 346 261 302  75 220
118 159 154 205 162 179 208 237 286 269 324 311 336 327 438 333 418 347 356 331 306 293  16 259
103  40 161 178 207 232 285 270 323 312 337 326 483 416 343 422 437 358 419 298 349 260 301  76
158 155 206 233 284 271 236 313 338 325 482 415 342 439 484 417 420 423 348 355 360 299 294  17
39 102 157 272 235 314 339 322 481 414 341 492 497 514 421 440 485 436 359 424 297 350  77 300
156 273 234 315 276 283 478 413 340 493 480 513 530 491 498 515 452 441 454 435 354 361  18 295
101  38 275 282 397 412 321 494 479 512 557 496 543 534 529 490 499 486 451 442 425 296 351  78
274 279 316 277 320 477 410 511 570 495 554 535 556 531 542 533 516 453 444 455 434 353 362  19
37 100 281 398 411 396 575 476 567 558 561 544 553 536 521 528 489 500 487 450 443 426  79 352
280 317 278 319 402 409 510 569 560 571 566 555 550 541 532 537 522 517 460 445 456 433  20 363
99  36 389 378 399 576 395 574 475 568 559 562 545 552 525 520 527 488 501 462 449 364 427  80
94 379 318 401 388 403 408 509 572 565 474 551 540 549 538 523 518 461 446 459 432 457 366  21
35  98  93 390 377 400 573 394 375 508 563 546 373 524 519 526 371 502 463 466 365 448  81 428
380  95 382 385 404 387 376 407 564 473 374 507 548 539 372 503 464 467 370 447 458 431  22 367
383  34  97  92 391  32 405  90 393  30 547  88 471  28 505  86 469  26 465  84 369  24 429  82
96 381 384  33 386  91 392  31 406  89 472  29 506  87 470  27 504  85 468  25 430  83 368  23

## ALGOL 68

Works with: ALGOL 68G version Any - tested with release 2.8.win32
# Non-recursive Knight's Tour with Warnsdorff's algorithm                #
# If there are multiple choices, backtrack if the first choice doesn't #
# find a solution #

# the size of the board #
INT board size = 8;

# directions for moves #
INT nne = 1, nee = 2, see = 3, sse = 4, ssw = 5, sww = 6, nww = 7, nnw = 8;

INT lowest move = nne;
INT highest move = nnw;

# the vertical position changes of the moves #
# nne, nee, see, sse, ssw, sww, nww, nnw #
[]INT offset v = ( -2, -1, 1, 2, 2, 1, -1, -2 );
# the horizontal position changes of the moves #
# nne, nee, see, sse, ssw, sww, nww, nnw #
[]INT offset h = ( 1, 2, 2, 1, -1, -2, -2, -1 );

MODE SQUARE = STRUCT( INT move # the number of the move that caused #
# the knight to reach this square #
, INT direction # the direction of the move that #
# brought the knight here - one of #
# nne, nee, see, sse, ssw, sww, nww #
# or nnw - used for backtracking #
# zero for the first move #
);

# the board #
[ board size, board size ]SQUARE board;

# initialises the board so there are no used squares #
PROC initialise board = VOID:
FOR row FROM 1 LWB board TO 1 UPB board
DO
FOR col FROM 2 LWB board TO 2 UPB board
DO
board[ row, col ] := ( 0, 0 )
OD
OD; # initialise board #

INT iterations := 0;
INT backtracks := 0;

# prints the board #
PROC print tour = VOID:
BEGIN

print( ( " a b c d e f g h", newline ) );
print( ( " +--------------------------------", newline ) );

FOR row FROM 1 UPB board BY -1 TO 1 LWB board
DO
print( ( whole( row, -3 ) ) );
print( ( "|" ) );

FOR col FROM 2 LWB board TO 2 UPB board
DO
print( ( " " ) );
print( ( whole( move OF board[ row, col ], -3 ) ) )
OD;
print( ( newline ) )
OD

END; # print tour #

# determines whether a move to the specified row and column is possible #
PROC can move to = ( INT row, INT col )BOOL:
IF row > 1 UPB board
OR row < 1 LWB board
OR col > 2 UPB board
OR col < 2 LWB board
THEN
# the position is not on the board #
FALSE
ELSE
# the move is legal, check the square is unoccupied #
move OF board[ row, col ] = 0
FI;

# used to hold counts of the number of moves that could be made in each #
# direction from the current square #
[ lowest move : highest move ]INT possible move count;

# sets the elements of possible move count to the number of moves that #
# could be made in each direction from the specified row and col #
PROC count moves in each direction from = ( INT row, INT col )VOID:
FOR move direction FROM lowest move TO highest move
DO

INT new row = row + offset v[ move direction ];
INT new col = col + offset h[ move direction ];

IF NOT can move to( new row, new col )
THEN
# can't move to this square #
possible move count[ move direction ] := -1
ELSE
# a move in this direction is possible #
# - count the number of moves that could be made from it #

possible move count[ move direction ] := 0;

FOR subsequent move FROM lowest move TO highest move
DO
IF can move to( new row + offset v[ subsequent move ]
, new col + offset h[ subsequent move ]
)
THEN
# have a possible subsequent move #
possible move count[ move direction ] +:= 1
FI
OD
FI

OD;

# update the board to the first knight's tour found starting from #
# "start row" and "start col". #
# return TRUE if one was found, FALSE otherwise #
PROC find tour = ( INT start row, INT start col )BOOL:
BEGIN

initialise board;

BOOL result := TRUE;

INT move number := 1;
INT row := start row;
INT col := start col;

# the tour will be complete when we have made as many moves #
# as there squares on the board #
INT final move = ( ( ( 1 UPB board ) + 1 ) - 1 LWB board )
* ( ( ( 2 UPB board ) + 1 ) - 2 LWB board )
;

# the first move is to place the knight on the starting square #
board[ row, col ] := ( move number, lowest move - 1 );
# start off with an unknown direction for the best move #
INT best direction := lowest move - 1;

# attempt to find a sequence of moves that will reach each square once #
WHILE
move number < final move AND result
DO

iterations +:= 1;

# count the number of moves possible from each possible move #
# from this square #
count moves in each direction from( row, col );

# find the direction with the lowest number of subsequent moves #

IF best direction < lowest move
THEN
# must find the best direction to move in #

INT lowest move count := highest move + 1;

FOR move direction FROM lowest move TO highest move
DO
IF possible move count[ move direction ] >= 0
AND possible move count[ move direction ] < lowest move count
THEN
# have a move with fewer possible subsequent moves #
best direction := move direction;
lowest move count := possible move count[ move direction ]
FI
OD

ELSE
# following a backtrack - find an alternative with the same #
# lowest number of possible moves - if there are any #
# if there aren't, we will backtrack again #

INT lowest move count := possible move count[ best direction ];

WHILE
best direction +:= 1;
IF best direction > highest move
THEN
# no more possible moves with the lowest number of #
# subsequent moves #
FALSE
ELSE
# keep looking if the number of moves from this square #
# isn't the lowest #
possible move count[ best direction ] /= lowest move count
FI
DO
SKIP
OD

FI;

IF best direction <= highest move
AND best direction >= lowest move
THEN
# we found a best possible move #

INT new row = row + offset v[ best direction ];
INT new col = col + offset h[ best direction ];

row := new row;
col := new col;
move number +:= 1;
board[ row, col ] := ( move number, best direction );

best direction := lowest move - 1

ELSE
# no more moves from this position - backtrack #

IF move number = 1
THEN
# at the starting position - no solution #
result := FALSE

ELSE
# not at the starting position - undo the latest move #

backtracks +:= 1;

move number -:= 1;

INT curr row := row;
INT curr col := col;

best direction := direction OF board[ curr row, curr col ];

row -:= offset v[ best direction ];
col -:= offset h[ best direction ];

# reset the square we just backtracked from #
board[ curr row, curr col ] := ( 0, 0 )

FI

FI

OD;

result
END; # find tour #

main:(

# get the starting position #

CHAR row;
CHAR col;

WHILE
print( ( "Enter starting row(1-8) and col(a-h): " ) );
read ( ( row, col, newline ) );
row < "1" OR row > "8" OR col < "a" OR col > "h"
DO
SKIP
OD;

# calculate the tour from that position, if possible #

IF find tour( ABS row - ABS "0", ( ABS col - ABS "a" ) + 1 )
THEN
# found a solution #
print tour
ELSE
# couldn't find a solution #
, ", backtracks: ", backtracks
, newline
)
)
FI

)
Output:
Enter starting row(1-8) and col(a-h): 5d
a   b   c   d   e   f   g   h
+--------------------------------
8|  51  18  53  20  41  44   3   6
7|  54  21  50  45   2   5  40  43
6|  17  52  19  58  49  42   7   4
5|  22  55  64   1  46  57  48  39
4|  33  16  23  56  59  38  29   8
3|  24  13  34  63  30  47  60  37
2|  15  32  11  26  35  62   9  28
1|  12  25  14  31  10  27  36  61

## ANSI Standard BASIC

Translation of: BBC BASIC

ANSI BASIC doesn't allow function parameters to be passed by reference so X and Y were made global variables.

100 DECLARE EXTERNAL FUNCTION choosemove
110 !
120 RANDOMIZE
130 PUBLIC NUMERIC X, Y, TRUE, FALSE
140 LET TRUE = -1
150 LET FALSE = 0
160 !
170 SET WINDOW 1,512,1,512
180 SET AREA COLOR "black"
190 FOR x=0 TO 512-128 STEP 128
200 FOR y=0 TO 512-128 STEP 128
210 PLOT AREA:x+64,y;x+128,y;x+128,y+64;x+64,y+64
220 PLOT AREA:x,y+64;x+64,y+64;x+64,y+128;x,y+128
230 NEXT y
240 NEXT x
250 !
260 SET LINE COLOR "red"
270 SET LINE WIDTH 6
280 !
290 PUBLIC NUMERIC Board(0 TO 7,0 TO 7)
300 LET X = 0
310 LET Y = 0
320 LET Total = 0
330 DO
340 LET Board(X,Y) = TRUE
350 PLOT LINES: X*64+32,Y*64+32;
360 LET Total = Total + 1
370 LOOP UNTIL choosemove(X, Y) = FALSE
380 IF Total <> 64 THEN STOP
390 END
400 !
410 EXTERNAL FUNCTION choosemove(X1, Y1)
420 DECLARE EXTERNAL SUB trymove
430 LET M = 9
440 CALL trymove(X1+1, Y1+2, M, newx, newy)
450 CALL trymove(X1+1, Y1-2, M, newx, newy)
460 CALL trymove(X1-1, Y1+2, M, newx, newy)
470 CALL trymove(X1-1, Y1-2, M, newx, newy)
480 CALL trymove(X1+2, Y1+1, M, newx, newy)
490 CALL trymove(X1+2, Y1-1, M, newx, newy)
500 CALL trymove(X1-2, Y1+1, M, newx, newy)
510 CALL trymove(X1-2, Y1-1, M, newx, newy)
520 IF M=9 THEN
530 LET choosemove = FALSE
540 EXIT FUNCTION
550 END IF
560 LET X = newx
570 LET Y = newy
580 LET choosemove = TRUE
590 END FUNCTION
600 !
610 EXTERNAL SUB trymove(X, Y, M, newx, newy)
620 !
630 DECLARE EXTERNAL FUNCTION validmove
640 IF validmove(X,Y) = 0 THEN EXIT SUB
650 IF validmove(X+1,Y+2) <> 0 THEN LET N = N + 1
660 IF validmove(X+1,Y-2) <> 0 THEN LET N = N + 1
670 IF validmove(X-1,Y+2) <> 0 THEN LET N = N + 1
680 IF validmove(X-1,Y-2) <> 0 THEN LET N = N + 1
690 IF validmove(X+2,Y+1) <> 0 THEN LET N = N + 1
700 IF validmove(X+2,Y-1) <> 0 THEN LET N = N + 1
710 IF validmove(X-2,Y+1) <> 0 THEN LET N = N + 1
720 IF validmove(X-2,Y-1) <> 0 THEN LET N = N + 1
730 IF N>M THEN EXIT SUB
740 IF N=M AND RND<.5 THEN EXIT SUB
750 LET M = N
760 LET newx = X
770 LET newy = Y
780 END SUB
790 !
800 EXTERNAL FUNCTION validmove(X,Y)
810 LET validmove = FALSE
820 IF X<0 OR X>7 OR Y<0 OR Y>7 THEN EXIT FUNCTION
830 IF Board(X,Y)=FALSE THEN LET validmove = TRUE
840 END FUNCTION

## AutoHotkey

Library: GDIP
#SingleInstance, Force
#NoEnv
SetBatchLines, -1
; Uncomment if Gdip.ahk is not in your standard library
;#Include, Gdip.ahk
If !pToken := Gdip_Startup(){
MsgBox, 48, Gdiplus error!, Gdiplus failed to start. Please ensure you have Gdiplus on your system.
ExitApp
}
; I've added a simple new function here, just to ensure if anyone is having any problems then to make sure they are using the correct library version
if (Gdip_LibraryVersion() < 1.30)
{
ExitApp
}
OnExit, Exit
tour := "a1 b3 d2 c4 a5 b7 d8 e6 d4 b5 c7 a8 b6 c8 a7 c6 b8 a6 b4 d5 e3 d1 b2 a4 c5 d7 f8 h7 f6 g8 h6 f7 h8 g6 e7 f5 h4 g2 e1 d3 e5 g4 f2 h1 g3 f1 h2 f3 g1 h3 g5 e4 d6 e8 g7 h5 f4 e2 c1 a2 c3 b1 a3 c2 "
; Knight's tour with maximum symmetry by George Jelliss, http://www.mayhematics.com/t/8f.htm
; I know, I know, but I followed the task outline to the letter! Besides, this path is the prettiest.

; Input: starting square
InputBox, start, Knight's Tour Start, Enter Knight's starting location in algebraic notation:, , , , , , , , b3
i := InStr(tour, start)
If i=0
{
}
; Output: move sequence
Msgbox % tour := SubStr(tour, i) . SubStr(tour, 1, i-1)

; Animation
tour .= SubStr(tour, 1, 3)
, CellSize := 30 ; pixels
, Width := Height := 9*CellSize
, TopLeftX := (A_ScreenWidth - Width) // 2
, TopLeftY := (A_ScreenHeight - Height) // 2
Gui, -Caption +E0x80000 +LastFound +AlwaysOnTop +ToolWindow +OwnDialogs
Gui, Show, NA ; show board (currently transparent)
hwnd1 := WinExist() ; required for Gdip
OnMessage(0x201, "WM_LBUTTONDOWN")
, hbm := CreateDIBSection(Width, Height)
, hdc := CreateCompatibleDC()
, obm := SelectObject(hdc, hbm)
, G := Gdip_GraphicsFromHDC(hdc)
, Gdip_SetSmoothingMode(G, 4)

Loop 1 ; remove '1' and uncomment next line to loop infinitely
{
;Gdip_GraphicsClear(G) ; uncomment to loop infinitely
cOdd := "0xFFFFCE9E" ; create brushes
, cEven := "0xFFD18B47"
, pBrushOdd := Gdip_BrushCreateSolid(cOdd)
, pBrushEven := Gdip_BrushCreateSolid(cEven)

Loop 64 ; layout board
{
Row := mod(A_Index-1,8)+1
, Col := (A_Index-1)//8+1
, Gdip_FillRectangle(G, mod(Row+Col,2) ? pBrushOdd : pBrushEven, Col * CellSize + 1, Row * CellSize + 1, CellSize - 2, CellSize - 2)
}
Gdip_DeleteBrush(pBrushOdd) ; cleanup memory
, Gdip_DeleteBrush(pBrushEven)
, UpdateLayeredWindow(hwnd1, hdc, TopLeftX, TopLeftY, Width, Height) ; update board

, pPen := Gdip_CreatePen(0x66FF0000, CellSize/10) ; create pen
, Algebraic := SubStr(tour,1,2) ; get starting coordinates
, x := (Asc(SubStr(Algebraic, 1, 1))-96+0.5)*CellSize
, y := (9.5-SubStr(Algebraic, 2, 1))*CellSize

Loop 64 ; trace path
{
Sleep, 0.5*1000
xold := x, yold := y ; a line has start and end points
, Algebraic := SubStr(tour,(A_Index)*3+1,2) ; get new coordinates
, x := (Asc(SubStr(Algebraic, 1, 1))-96+0.5)*CellSize
, y := (9.5-SubStr(Algebraic, 2, 1))*CellSize
, Gdip_DrawLine(G, pPen, xold, yold, x, y)
, UpdateLayeredWindow(hwnd1, hdc, TopLeftX, TopLeftY, Width, Height) ; update board
}
Gdip_DeletePen(pPen)
}
Return

GuiEscape:
ExitApp

Exit:
Gdip_Shutdown(pToken)
ExitApp

WM_LBUTTONDOWN()
{
If (A_Gui = 1)
PostMessage, 0xA1, 2
}
Output:

For start at b3

b3 d2 c4 a5 b7 d8 e6 d4 b5 c7 a8 b6 c8 a7 c6 b8 a6 b4 d5 e3 d1 b2 a4 c5 d7 f8 h7 f6 g8 h6 f7 h8 g6 e7 f5 h4 g2 e1 d3 e5 g4 f2 h1 g3 f1 h2 f3 g1 h3 g5 e4 d6 e8 g7 h5 f4 e2 c1 a2 c3 b1 a3 c2 a1

... plus an animation.

## AWK

# syntax: GAWK -f KNIGHTS_TOUR.AWK [-v sr=x] [-v sc=x]
#
# examples:
# GAWK -f KNIGHTS_TOUR.AWK (default)
# GAWK -f KNIGHTS_TOUR.AWK -v sr=1 -v sc=1 start at top left (default)
# GAWK -f KNIGHTS_TOUR.AWK -v sr=1 -v sc=8 start at top right
# GAWK -f KNIGHTS_TOUR.AWK -v sr=8 -v sc=8 start at bottom right
# GAWK -f KNIGHTS_TOUR.AWK -v sr=8 -v sc=1 start at bottom left
#
BEGIN {
N = 8 # board size
if (sr == "") { sr = 1 } # starting row
if (sc == "") { sc = 1 } # starting column
split("2 2 -2 -2 1 1 -1 -1",X," ")
split("1 -1 1 -1 2 -2 2 -2",Y," ")
printf("\n%dx%d board: starting row=%d col=%d\n",N,N,sr,sc)
move(sr,sc,0)
exit(1)
}
function move(x,y,m) {
if (cantMove(x,y)) {
return(0)
}
P[x,y] = ++m
if (m == N ^ 2) {
printBoard()
exit(0)
}
tryBestMove(x,y,m)
}
function cantMove(x,y) {
return( P[x,y] || x<1 || x>N || y<1 || y>N )
}
function tryBestMove(x,y,m, i) {
i = bestMove(x,y)
move(x+X[i],y+Y[i],m)
}
function bestMove(x,y, arg1,arg2,c,i,min,out) {
# Warnsdorff's rule: go to where there are fewest next moves
min = N ^ 2 + 1
for (i in X) {
arg1 = x + X[i]
arg2 = y + Y[i]
if (!cantMove(arg1,arg2)) {
c = countNext(arg1,arg2)
if (c < min) {
min = c
out = i
}
}
}
return(out)
}
function countNext(x,y, i,out) {
for (i in X) {
out += (!cantMove(x+X[i],y+Y[i]))
}
return(out)
}
function printBoard( i,j,leng) {
leng = length(N*N)
for (i=1; i<=N; i++) {
for (j=1; j<=N; j++) {
printf(" %*d",leng,P[i,j])
}
printf("\n")
}
}

output:

8x8 board: starting row=1 col=1
1 50 15 32 61 28 13 30
16 33 64 55 14 31 60 27
51  2 49 44 57 62 29 12
34 17 56 63 54 47 26 59
3 52 45 48 43 58 11 40
18 35 20 53 46 41  8 25
21  4 37 42 23  6 39 10
36 19 22  5 38  9 24  7

## BBC BASIC

VDU 23,22,256;256;16,16,16,128
VDU 23,23,4;0;0;0;
OFF
GCOL 4,15
FOR x% = 0 TO 512-128 STEP 128
RECTANGLE FILL x%,0,64,512
NEXT
FOR y% = 0 TO 512-128 STEP 128
RECTANGLE FILL 0,y%,512,64
NEXT
GCOL 9

DIM Board%(7,7)
X% = 0
Y% = 0
Total% = 0
REPEAT
Board%(X%,Y%) = TRUE
IF Total% DRAW X%*64+32,Y%*64+32 ELSE MOVE X%*64+32,Y%*64+32
Total% += 1
UNTIL NOT FNchoosemove(X%, Y%)
IF Total%<>64 STOP
REPEAT WAIT 1 : UNTIL FALSE
END

DEF FNchoosemove(RETURN X%, RETURN Y%)
LOCAL M%, newx%, newy%
M% = 9
PROCtrymove(X%+1, Y%+2, M%, newx%, newy%)
PROCtrymove(X%+1, Y%-2, M%, newx%, newy%)
PROCtrymove(X%-1, Y%+2, M%, newx%, newy%)
PROCtrymove(X%-1, Y%-2, M%, newx%, newy%)
PROCtrymove(X%+2, Y%+1, M%, newx%, newy%)
PROCtrymove(X%+2, Y%-1, M%, newx%, newy%)
PROCtrymove(X%-2, Y%+1, M%, newx%, newy%)
PROCtrymove(X%-2, Y%-1, M%, newx%, newy%)
IF M%=9 THEN = FALSE
X% = newx% : Y% = newy%
= TRUE

DEF PROCtrymove(X%, Y%, RETURN M%, RETURN newx%, RETURN newy%)
LOCAL N%
IF NOT FNvalidmove(X%,Y%) THEN ENDPROC
IF FNvalidmove(X%+1,Y%+2) N% += 1
IF FNvalidmove(X%+1,Y%-2) N% += 1
IF FNvalidmove(X%-1,Y%+2) N% += 1
IF FNvalidmove(X%-1,Y%-2) N% += 1
IF FNvalidmove(X%+2,Y%+1) N% += 1
IF FNvalidmove(X%+2,Y%-1) N% += 1
IF FNvalidmove(X%-2,Y%+1) N% += 1
IF FNvalidmove(X%-2,Y%-1) N% += 1
IF N%>M% THEN ENDPROC
IF N%=M% IF RND(2)=1 THEN ENDPROC
M% = N%
newx% = X% : newy% = Y%
ENDPROC

DEF FNvalidmove(X%,Y%)
IF X%<0 OR X%>7 OR Y%<0 OR Y%>7 THEN = FALSE
= NOT(Board%(X%,Y%))

## Bracmat

( knightsTour
= validmoves WarnsdorffSort algebraicNotation init solve
, x y fieldsToVisit
. ~
| ( validmoves
= x y jumps moves
.  !arg:(?x.?y)
& :?moves
& ( jumps
= dx dy Fs fxs fys fx fy
.  !arg:(?dx.?dy)
& 1 -1:?Fs
& !Fs:?fxs
& whl
' ( !fxs:%?fx ?fxs
& !Fs:?fys
& whl
' ( !fys:%?fy ?fys
& ( (!x+!fx*!dx.!y+!fy*!dy)
: (>0:<9.>0:<9)
|
)
!moves
: ?moves
)
)
)
& jumps\$(1.2)
& jumps\$(2.1)
& !moves
)
& ( init
= fields x y
.  :?fields
& 0:?x
& whl
' ( 1+!x:<9:?x
& 0:?y
& whl
' ( 1+!y:<9:?y
& (!x.!y) !fields:?fields
)
)
& !fields
)
& init\$:?fieldsToVisit
& ( WarnsdorffSort
= sum moves elm weightedTerms
. ( weightedTerms
= pos alts fieldsToVisit moves move weight
.  !arg:(%?pos ?alts.?fieldsToVisit)
& (  !fieldsToVisit:!pos
& (0.!pos)
|  !fieldsToVisit:? !pos ?
& validmoves\$!pos:?moves
& 0:?weight
& whl
' ( !moves:%?move ?moves
& (  !fieldsToVisit:? !move ?
& !weight+1:?weight
|
)
)
& (!weight.!pos)
| 0
)
+ weightedTerms\$(!alts.!fieldsToVisit)
| 0
)
& weightedTerms\$!arg:?sum
& :?moves
& whl
' ( !sum:(#.?elm)+?sum
& !moves !elm:?moves
)
& !moves
)
& ( solve
= pos alts fieldsToVisit A Z tailOfSolution
.  !arg:(%?pos ?alts.?fieldsToVisit)
& (  !fieldsToVisit:?A !pos ?Z
& ( !A !Z:&
| solve
\$ ( WarnsdorffSort\$(validmoves\$!pos.!A !Z)
. !A !Z
)
)
| solve\$(!alts.!fieldsToVisit)
)
: ?tailOfSolution
& !pos !tailOfSolution
)
& ( algebraicNotation
= x y
.  !arg:(?x.?y) ?arg
& str\$(chr\$(asc\$a+!x+-1) !y " ")
algebraicNotation\$!arg
|
)
& @(!arg:?x #?y)
& asc\$!x+-1*asc\$a+1:?x
& str
\$ (algebraicNotation\$(solve\$((!x.!y).!fieldsToVisit)))
)
& out\$(knightsTour\$a1);
a1 b3 a5 b7 d8 f7 h8 g6 f8 h7 g5 h3 g1 e2 c1 a2 b4 a6 b8 c6 a7 c8 e7 g8 h6 g4 h2 f1 d2 b1 a3 c2 e1 f3 h4 g2 e3 d1 b2 a4 c3 b5 d4 f5 d6 c4 e5 d3 f2 h1 g3 e4 c5 d7 b6 a8 c7 d5 f4 e6 g7 e8 f6 h5

## C

For an animated version using OpenGL, see Knight's tour/C.

The following draws on console the progress of the horsie. Specify board size on commandline, or use default 8.

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <unistd.h>

typedef unsigned char cell;
int dx[] = { -2, -2, -1, 1, 2, 2, 1, -1 };
int dy[] = { -1, 1, 2, 2, 1, -1, -2, -2 };

void init_board(int w, int h, cell **a, cell **b)
{
int i, j, k, x, y, p = w + 4, q = h + 4;
/* b is board; a is board with 2 rows padded at each side */
a[0] = (cell*)(a + q);
b[0] = a[0] + 2;

for (i = 1; i < q; i++) {
a[i] = a[i-1] + p;
b[i] = a[i] + 2;
}

memset(a[0], 255, p * q);
for (i = 0; i < h; i++) {
for (j = 0; j < w; j++) {
for (k = 0; k < 8; k++) {
x = j + dx[k], y = i + dy[k];
if (b[i+2][j] == 255) b[i+2][j] = 0;
b[i+2][j] += x >= 0 && x < w && y >= 0 && y < h;
}
}
}
}

#define E "\033["
int walk_board(int w, int h, int x, int y, cell **b)
{
int i, nx, ny, least;
int steps = 0;
printf(E"H"E"J"E"%d;%dH"E"32m[]"E"m", y + 1, 1 + 2 * x);

while (1) {
/* occupy cell */
b[y][x] = 255;

/* reduce all neighbors' neighbor count */
for (i = 0; i < 8; i++)
b[ y + dy[i] ][ x + dx[i] ]--;

/* find neighbor with lowest neighbor count */
least = 255;
for (i = 0; i < 8; i++) {
if (b[ y + dy[i] ][ x + dx[i] ] < least) {
nx = x + dx[i];
ny = y + dy[i];
least = b[ny][nx];
}
}

if (least > 7) {
printf(E"%dH", h + 2);
return steps == w * h - 1;
}

if (steps++) printf(E"%d;%dH[]", y + 1, 1 + 2 * x);
x = nx, y = ny;
printf(E"%d;%dH"E"31m[]"E"m", y + 1, 1 + 2 * x);
fflush(stdout);
usleep(120000);
}
}

int solve(int w, int h)
{
int x = 0, y = 0;
cell **a, **b;
a = malloc((w + 4) * (h + 4) + sizeof(cell*) * (h + 4));
b = malloc((h + 4) * sizeof(cell*));

while (1) {
init_board(w, h, a, b);
if (walk_board(w, h, x, y, b + 2)) {
printf("Success!\n");
return 1;
}
if (++x >= w) x = 0, y++;
if (y >= h) {
printf("Failed to find a solution\n");
return 0;
}
printf("Any key to try next start position");
getchar();
}
}

int main(int c, char **v)
{
int w, h;
if (c < 2 || (w = atoi(v[1])) <= 0) w = 8;
if (c < 3 || (h = atoi(v[2])) <= 0) h = w;
solve(w, h);

return 0;
}

## C#

using System;
using System.Collections.Generic;

namespace prog
{
class MainClass
{
const int N = 8;

readonly static int[,] moves = { {+1,-2},{+2,-1},{+2,+1},{+1,+2},
{-1,+2},{-2,+1},{-2,-1},{-1,-2} };
struct ListMoves
{
public int x, y;
public ListMoves( int _x, int _y ) { x = _x; y = _y; }
}

public static void Main (string[] args)
{
int[,] board = new int[N,N];
board.Initialize();

int x = 0, // starting position
y = 0;

List<ListMoves> list = new List<ListMoves>(N*N);

do
{
if ( Move_Possible( board, x, y ) )
{
int move = board[x,y];
board[x,y]++;
x += moves[move,0];
y += moves[move,1];
}
else
{
if ( board[x,y] >= 8 )
{
board[x,y] = 0;
list.RemoveAt(list.Count-1);
if ( list.Count == 0 )
{
Console.WriteLine( "No solution found." );
return;
}
x = list[list.Count-1].x;
y = list[list.Count-1].y;
}
board[x,y]++;
}
}
while( list.Count < N*N );

int last_x = list[0].x,
last_y = list[0].y;
string letters = "ABCDEFGH";
for( int i=1; i<list.Count; i++ )
{
Console.WriteLine( string.Format("{0,2}: ", i) + letters[last_x] + (last_y+1) + " - " + letters[list[i].x] + (list[i].y+1) );

last_x = list[i].x;
last_y = list[i].y;
}
}

static bool Move_Possible( int[,] board, int cur_x, int cur_y )
{
if ( board[cur_x,cur_y] >= 8 )
return false;

int new_x = cur_x + moves[board[cur_x,cur_y],0],
new_y = cur_y + moves[board[cur_x,cur_y],1];

if ( new_x >= 0 && new_x < N && new_y >= 0 && new_y < N && board[new_x,new_y] == 0 )
return true;

return false;
}
}
}

## C++

Works with: C++11

Uses Warnsdorff's rule and (iterative) backtracking if that fails.

#include <iostream>
#include <iomanip>
#include <array>
#include <string>
#include <tuple>
#include <algorithm>
using namespace std;

template<int N = 8>
class Board
{
public:
array<pair<int, int>, 8> moves;
array<array<int, N>, N> data;

Board()
{
moves[0] = make_pair(2, 1);
moves[1] = make_pair(1, 2);
moves[2] = make_pair(-1, 2);
moves[3] = make_pair(-2, 1);
moves[4] = make_pair(-2, -1);
moves[5] = make_pair(-1, -2);
moves[6] = make_pair(1, -2);
moves[7] = make_pair(2, -1);
}

array<int, 8> sortMoves(int x, int y) const
{
array<tuple<int, int>, 8> counts;
for(int i = 0; i < 8; ++i)
{
int dx = get<0>(moves[i]);
int dy = get<1>(moves[i]);

int c = 0;
for(int j = 0; j < 8; ++j)
{
int x2 = x + dx + get<0>(moves[j]);
int y2 = y + dy + get<1>(moves[j]);

if (x2 < 0 || x2 >= N || y2 < 0 || y2 >= N)
continue;
if(data[y2][x2] != 0)
continue;

c++;
}

counts[i] = make_tuple(c, i);
}

// Shuffle to randomly break ties
random_shuffle(counts.begin(), counts.end());

// Lexicographic sort
sort(counts.begin(), counts.end());

array<int, 8> out;
for(int i = 0; i < 8; ++i)
out[i] = get<1>(counts[i]);
return out;
}

void solve(string start)
{
for(int v = 0; v < N; ++v)
for(int u = 0; u < N; ++u)
data[v][u] = 0;

int x0 = start[0] - 'a';
int y0 = N - (start[1] - '0');
data[y0][x0] = 1;

array<tuple<int, int, int, array<int, 8>>, N*N> order;
order[0] = make_tuple(x0, y0, 0, sortMoves(x0, y0));

int n = 0;
while(n < N*N-1)
{
int x = get<0>(order[n]);
int y = get<1>(order[n]);

bool ok = false;
for(int i = get<2>(order[n]); i < 8; ++i)
{
int dx = moves[get<3>(order[n])[i]].first;
int dy = moves[get<3>(order[n])[i]].second;

if(x+dx < 0 || x+dx >= N || y+dy < 0 || y+dy >= N)
continue;
if(data[y + dy][x + dx] != 0)
continue;

get<2>(order[n]) = i + 1;
++n;
data[y+dy][x+dx] = n + 1;
order[n] = make_tuple(x+dx, y+dy, 0, sortMoves(x+dx, y+dy));
ok = true;
break;
}

if(!ok) // Failed. Backtrack.
{
data[y][x] = 0;
--n;
}
}
}

template<int N>
friend ostream& operator<<(ostream &out, const Board<N> &b);
};

template<int N>
ostream& operator<<(ostream &out, const Board<N> &b)
{
for (int v = 0; v < N; ++v)
{
for (int u = 0; u < N; ++u)
{
if (u != 0) out << ",";
out << setw(3) << b.data[v][u];
}
out << endl;
}
return out;
}

int main()
{
Board<5> b1;
b1.solve("c3");
cout << b1 << endl;

Board<8> b2;
b2.solve("b5");
cout << b2 << endl;

Board<31> b3; // Max size for <1000 squares
b3.solve("a1");
cout << b3 << endl;
return 0;
}

Output:

23, 16, 11,  6, 21
10,  5, 22, 17, 12
15, 24,  1, 20,  7
4,  9, 18, 13,  2
25, 14,  3,  8, 19

63, 20,  3, 24, 59, 36,  5, 26
2, 23, 64, 37,  4, 25, 58, 35
19, 62, 21, 50, 55, 60, 27,  6
22,  1, 54, 61, 38, 45, 34, 57
53, 18, 49, 44, 51, 56,  7, 28
12, 15, 52, 39, 46, 31, 42, 33
17, 48, 13, 10, 43, 40, 29,  8
14, 11, 16, 47, 30,  9, 32, 41

275,112, 19,116,277,604, 21,118,823,770, 23,120,961,940, 25,122,943,926, 27,124,917,898, 29,126,911,872, 31,128,197,870, 33
18,115,276,601, 20,117,772,767, 22,119,958,851, 24,121,954,941, 26,123,936,925, 28,125,912,899, 30,127,910,871, 32,129,198
111,274,113,278,605,760,603,822,771,824,769,948,957,960,939,944,953,942,927,916,929,918,897,908,913,900,873,196,875, 34,869
114, 17,600,273,602,775,766,773,768,949,850,959,852,947,952,955,932,937,930,935,924,915,920,905,894,909,882,901,868,199,130
271,110,279,606,759,610,761,776,821,764,825,816,951,956,853,938,945,934,923,928,919,896,893,914,907,904,867,874,195,876, 35
16,581,272,599,280,607,774,765,762,779,950,849,826,815,946,933,854,931,844,857,890,921,906,895,886,883,902,881,200,131,194
109,270,281,580,609,758,611,744,777,820,763,780,817,848,827,808,811,846,855,922,843,858,889,892,903,866,885,192,877, 36,201
282, 15,582,269,598,579,608,757,688,745,778,819,754,783,814,847,828,807,810,845,856,891,842,859,884,887,880,863,202,193,132
267,108,283,578,583,612,689,614,743,756,691,746,781,818,753,784,809,812,829,806,801,840,835,888,865,862,203,878,191,530, 37
14,569,268,585,284,597,576,619,690,687,742,755,692,747,782,813,752,785,802,793,830,805,860,841,836,879,864,529,204,133,190
107,266,285,570,577,584,613,686,615,620,695,684,741,732,711,748,739,794,751,786,803,800,839,834,861,528,837,188,531, 38,205
286, 13,568,265,586,575,596,591,618,685,616,655,696,693,740,733,712,749,738,795,792,831,804,799,838,833,722,527,206,189,134
263,106,287,508,571,590,587,574,621,592,639,694,683,656,731,710,715,734,787,750,737,796,791,832,721,798,207,532,187,474, 39
12,417,264,567,288,509,572,595,588,617,654,657,640,697,680,713,730,709,716,735,788,727,720,797,790,723,526,473,208,135,186
105,262,289,416,507,566,589,512,573,622,593,638,653,682,659,698,679,714,729,708,717,736,789,726,719,472,533,184,475, 40,209
290, 11,418,261,502,415,510,565,594,513,562,641,658,637,652,681,660,699,678,669,728,707,718,675,724,525,704,471,210,185,136
259,104,291,414,419,506,503,514,511,564,623,548,561,642,551,636,651,670,661,700,677,674,725,706,703,534,211,476,183,396, 41
10,331,260,493,292,501,420,495,504,515,498,563,624,549,560,643,662,635,650,671,668,701,676,673,524,705,470,395,212,137,182
103,258,293,330,413,494,505,500,455,496,547,516,485,552,625,550,559,644,663,634,649,672,667,702,535,394,477,180,397, 42,213
294,  9,332,257,492,329,456,421,490,499,458,497,546,517,484,553,626,543,558,645,664,633,648,523,666,469,536,393,220,181,138
255,102,295,328,333,412,491,438,457,454,489,440,459,486,545,518,483,554,627,542,557,646,665,632,537,478,221,398,179,214, 43
8,319,256,335,296,345,326,409,422,439,436,453,488,441,460,451,544,519,482,555,628,541,522,647,468,631,392,219,222,139,178
101,254,297,320,327,334,411,346,437,408,423,368,435,452,487,442,461,450,445,520,481,556,629,538,479,466,399,176,215, 44,165
298,  7,318,253,336,325,344,349,410,347,360,407,424,383,434,427,446,443,462,449,540,521,480,467,630,391,218,223,164,177,140
251,100,303,300,321,316,337,324,343,350,369,382,367,406,425,384,433,428,447,444,463,430,539,390,465,400,175,216,169,166, 45
6,299,252,317,304,301,322,315,348,361,342,359,370,381,366,405,426,385,432,429,448,389,464,401,174,217,224,163,150,141,168
99,250,241,302,235,248,307,338,323,314,351,362,341,358,371,380,365,404,377,386,431,402,173,388,225,160,153,170,167, 46,143
240,  5, 98,249,242,305,234,247,308,339,232,313,352,363,230,357,372,379,228,403,376,387,226,159,154,171,162,149,142,151, 82
63,  2,239, 66, 97,236,243,306,233,246,309,340,231,312,353,364,229,356,373,378,227,158,375,172,161,148,155,152, 83,144, 47
4, 67, 64, 61,238, 69, 96, 59,244, 71, 94, 57,310, 73, 92, 55,354, 75, 90, 53,374, 77, 88, 51,156, 79, 86, 49,146, 81, 84
1, 62,  3, 68, 65, 60,237, 70, 95, 58,245, 72, 93, 56,311, 74, 91, 54,355, 76, 89, 52,157, 78, 87, 50,147, 80, 85, 48,145

## Common Lisp

Works with: clisp version 2.49

This interactive program will ask for a starting case in algebraic notation and, also, whether a closed tour is desired. Each next move is selected according to Warnsdorff's rule; ties are broken at random.

The closed tour algorithm is quite crude: just find tours over and over until one happens to be closed by chance.

This code is quite verbose: I tried to make it easy for myself and for other to follow and understand. I'm not a Lisp expert, so I probably missed some idiomatic shortcuts I could have used to make it shorter.

For some reason, the interactive part does not work with sbcl, but it works fine wit clisp.

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; Solving the knight's tour.  ;;;
;;; Warnsdorff's rule with random tie break.  ;;;
;;; Optionally outputs a closed tour.  ;;;
;;; Options from interactive prompt.  ;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

(defparameter *side* 8)

(defun generate-chessboard (n)
(loop for i below n append
(loop for j below n collect (complex i j))))

(defparameter *chessboard*
(generate-chessboard *side*))

(defun complex->algebraic (n)
;; returns a string like "b2"
(concatenate 'string
;; 'a' is char #97: add it to the offset
(string (character (+ 97 (realpart n))))
;; indices start at 0, but algebraic starts at 1
(string (digit-char (+ 1 (imagpart n))))))

(defun algebraic->complex (string)
;; takes a string like "e4"
(let ((row (char string 0))
(col (char string 1)))
(complex (- (char-code row) 97)
(- (digit-char-p col) 1))))

(defconstant *knight-directions*
(list
(complex 1 2)
(complex 2 1)
(complex 1 -2)
(complex 2 -1)
(complex -1 2)
(complex -2 1)
(complex -1 -2)
(complex -2 -1)))

(defun find-legal-moves (moves-list)
;; 2. the move must not be on a case already visited
(remove-if (lambda (m) (member m moves-list))
;; 1. the move must be within the chessboard
(intersection
(mapcar (lambda (i) (+ (car moves-list) i)) *knight-directions*)
*chessboard*)))

;; Select between two moves by Warnsdorff's rule:
;; pick the one with the lowest index or else
;; randomly break the tie.
;; Takes a cons in the form (n . #C(x y)).
;; This will be the sorting rule for picking the next move.
(defun w-rule (a b)
(cond ((< (car a) (car b)) t)
((> (car a) (car b)) nil)
((= (car a) (car b))
(zerop (random 2)))))

;; For every legal move in a given position,
;; look forward one move and return a cons
;; in the form (n . #C(x y)) where n is
;; how many next free moves follow the first move.
(defun return-weighted-moves (moves)
(let ((candidates (find-legal-moves moves)))
(loop for mv in candidates collect
(cons
(list-length (find-legal-moves (cons mv moves)))
mv))))

;; Given a list of weighted moves (as above),
;; pick one according to the w-rule
(defun pick-among-weighted-moves (moves)
;; prune dead ends one move early
(let ((possible-moves
(remove-if (lambda(m) (zerop (car m))) moves)))
(cdar (sort possible-moves #'w-rule))))

(defun make-move (moves-list)
(let ((next-move
(if (< (list-length moves-list) (1- (list-length *chessboard*)))
(pick-among-weighted-moves (return-weighted-moves moves-list))
(car (find-legal-moves moves-list)))))
(cons next-move moves-list)))

(defun make-tour (moves-list)
;; takes a list of moves as an argument
(make-tour (last moves-list))
(if (= (list-length moves-list) (list-length *chessboard*))
moves-list
(make-tour (make-move moves-list)))))

(defun make-closed-tour (moves-list)
(let ((tour (make-tour moves-list)))
(if (tour-closed-p tour)
tour
(make-closed-tour moves-list))))

(defun tour-closed-p (tour)
;; takes a full tour as an argument
(let ((start (car (last tour)))
(end (car tour)))
;; is the first position a legal move, when
;; viewed from the last move?
(if (member start (find-legal-moves (list end))) ; find-legal-moves takes a list
t nil)))

(defun print-tour-linear (tour)
;; takes a tour (moves list) with the last move first
;; and prints it nicely in algebraic notation
(let ((moves (mapcar #'complex->algebraic (reverse tour))))
(format t "~{~A~^ -> ~}" moves)))

(defun tour->matrix (tour)
;; takes a tour and makes a row-by-row 2D matrix
;; from top to bottom (for further formatting & printing)
(flet ((index-tour (tour) ; 1st local function
(loop for i below (length tour)
;; starting from index 1, not 0, so add 1;
;; reverse because the last move is still in the car
collect (cons (nth i (reverse tour)) (1+ i))))
(get-row (n tour) ; 2nd local function
;; in every row, the imaginary part (vertical offset) stays the same
(remove-if-not (lambda (e) (= n (imagpart (car e)))) tour)))
(let* ((indexed-tour (index-tour tour))
(ordered-indexed-tour
;; make a list of ordered rows
(loop for i from (1- *side*) downto 0 collect
(sort (get-row i indexed-tour)
(lambda (a b) (< (realpart (car a)) (realpart (car b))))))))
;; clean up, leaving only the indices
(mapcar (lambda (e) (mapcar #'cdr e)) ordered-indexed-tour))))

(defun print-tour-matrix (tour)
(mapcar (lambda (row)
(format t "~{~3d~}~&" row)) (tour->matrix tour)))

;;; Handling options

(defstruct options
closed
start
grid)

(defparameter *opts* (make-options))

;;; Interactive part

(defun prompt()
(format t "Starting case (leave blank for random)? ")
(if (member start (mapcar #'complex->algebraic *chessboard*) :test #'equal)
(setf (options-start *opts*) start))
(format t "Require a closed tour (yes or default to no)? ")
(if (or (equal closed "y") (equal closed "yes"))
(setf (options-closed *opts*) t)))))

(defun main ()
(let* ((start
(if (options-start *opts*)
(algebraic->complex (options-start *opts*))
(complex (random *side*) (random *side*))))
(closed (options-closed *opts*))
(tour
(if closed
(make-closed-tour (list start))
(make-tour (list start)))))
(fresh-line)
(if closed (princ "Closed "))
(princ "Knight's tour")
(if (options-start *opts*)
(princ ":")
(princ " (starting on a random case):"))
(fresh-line)
(print-tour-linear tour)
(princ #\newline)
(princ #\newline)
(print-tour-matrix tour)))

;;; Good to go: invocation!

(prompt)
(main)
Output:
Starting case (leave blank for random)? a8
Require a closed tour (yes or default to no)? y

Closed Knight's tour:
a8 -> c7 -> e8 -> g7 -> h5 -> g3 -> h1 -> f2 -> h3 -> g1 -> e2 -> c1 -> a2 -> b4 -> a6 -> b8 -> d7 -> f8 -> h7 -> g5 -> e6 -> d8 -> b7 -> a5 -> b3 -> a1 -> c2 -> e1 -> g2 -> f4 -> d3 -> c5 -> a4 -> b2 -> d1 -> c3 -> b1 -> a3 -> b5 -> a7 -> c6 -> d4 -> f3 -> h4 -> g6 -> h8 -> f7 -> e5 -> g4 -> h2 -> f1 -> d2 -> e4 -> f6 -> g8 -> h6 -> f5 -> e7 -> d5 -> e3 -> c4 -> d6 -> c8 -> b6

1 16 63 22  3 18 55 46
40 23  2 17 58 47  4 19
15 64 41 62 21 54 45 56
24 39 32 59 48 57 20  5
33 14 61 42 53 30 49 44
38 25 36 31 60 43  6  9
13 34 27 52 11  8 29 50
26 37 12 35 28 51 10  7

## Clojure

Using warnsdorff's rule

(defn isin? [x li]
(not= [] (filter #(= x %) li)))

(defn options [movements pmoves n]
(let [x (first (last movements)) y (second (last movements))
op (vec (map #(vector (+ x (first %)) (+ y (second %))) pmoves))
vop (filter #(and (>= (first %) 0) (>= (last %) 0)) op)
vop1 (filter #(and (< (first %) n) (< (last %) n)) vop)]
(vec (filter #(not (isin? % movements)) vop1))))

(defn next-move [movements pmoves n]
(let [op (options movements pmoves n)
sp (map #(vector % (count (options (conj movements %) pmoves n))) op)
m (apply min (map last sp))]
(first (rand-nth (filter #(= m (last %)) sp)))))

(defn jumps [n pos]
(let [movements (vector pos)
pmoves [[1 2] [1 -2] [2 1] [2 -1]
[-1 2] [-1 -2] [-2 -1] [-2 1]]]
(loop [mov movements x 1]
(if (= x (* n n))
mov
(let [np (next-move mov pmoves n)]
(recur (conj mov np) (inc x)))))))

Output:
(jumps 5 [0 0])
[[0 0] [1 2] [0 4] [2 3] [4 4] [3 2] [4 0] [2 1] [1 3] [0 1] [2 0] [4 1] [3 3] [1 4] [0 2] [1 0] [3 1] [4 3] [2 4] [0 3] [1 1] [3 0] [4 2] [3 4] [2 2]]

(jumps 8 [0 0])
[[0 0] [2 1] [4 0] [6 1] [7 3] [6 5] [7 7] [5 6] [3 7] [1 6] [0 4] [1 2] [2 0] [0 1] [1 3] [0 5] [1 7] [2 5] [0 6] [2 7] [4 6] [6 7] [7 5] [6 3] [7 1] [5 0] [3 1] [1 0] [0 2] [1 4] [3 5] [4 7] [6 6] [7 4] [6 2] [7 0] [5 1] [7 2] [6 0] [4 1] [5 3] [3 2] [4 4] [5 2] [3 3] [5 4] [4 2] [2 3] [1 1] [3 0] [2 2] [0 3] [2 4] [4 3] [6 4] [4 5] [2 6] [0 7] [1 5] [3 4] [5 5] [7 6] [5 7] [3 6]]

(let [j (jumps 40 [0 0])]        ;; are
(and (distinct? j)             ;; all squares only once? and
(= (count j) (* 40 40)))) ;; 40*40 squares?
true

## CoffeeScript

This algorithm finds 100,000 distinct solutions to the 8x8 problem in about 30 seconds. It precomputes knight moves up front, so it turns into a pure graph traversal problem. The program uses iteration and backtracking to find solutions.

graph_tours = (graph, max_num_solutions) ->
# graph is an array of arrays
# graph[3] = [4, 5] means nodes 4 and 5 are reachable from node 3
#
# Returns an array of tours (up to max_num_solutions in size), where
# each tour is an array of nodes visited in order, and where each
# tour visits every node in the graph exactly once.
#
complete_tours = []
visited = (false for node in graph)
dead_ends = ({} for node in graph)
tour = [0]

valid_neighbors = (i) ->
arr = []
for neighbor in graph[i]
continue if visited[neighbor]
arr.push neighbor
arr

next_square_to_visit = (i) ->
arr = valid_neighbors i
return null if arr.length == 0

# We traverse to our neighbor who has the fewest neighbors itself.
fewest_neighbors = valid_neighbors(arr[0]).length
neighbor = arr[0]
for i in [1...arr.length]
n = valid_neighbors(arr[i]).length
if n < fewest_neighbors
fewest_neighbors = n
neighbor = arr[i]
neighbor

while tour.length > 0
current_square = tour[tour.length - 1]
visited[current_square] = true
next_square = next_square_to_visit current_square
if next_square?
tour.push next_square
if tour.length == graph.length
complete_tours.push (n for n in tour) # clone
break if complete_tours.length == max_num_solutions
# pessimistically call this a dead end
current_square = next_square
else
# we backtrack
doomed_square = tour.pop()
visited[doomed_square] = false
complete_tours

knight_graph = (board_width) ->
# Turn the Knight's Tour into a pure graph-traversal problem
# by precomputing all the legal moves. Returns an array of arrays,
# where each element in any subarray is the index of a reachable node.
index = (i, j) ->
# index squares from 0 to n*n - 1
board_width * i + j

reachable_squares = (i, j) ->
deltas = [
[ 1, 2]
[ 1, -2]
[ 2, 1]
[ 2, -1]
[-1, 2]
[-1, -2]
[-2, 1]
[-2, -1]
]
neighbors = []
for delta in deltas
[di, dj] = delta
ii = i + di
jj = j + dj
if 0 <= ii < board_width
if 0 <= jj < board_width
neighbors.push index(ii, jj)
neighbors

graph = []
for i in [0...board_width]
for j in [0...board_width]
graph[index(i, j)] = reachable_squares i, j
graph

illustrate_knights_tour = (tour, board_width) ->
return " _" if !n?
return " " + n if n < 10
"#{n}"

console.log "\n------"
moves = {}
for square, i in tour
moves[square] = i + 1
for i in [0...board_width]
s = ''
for j in [0...board_width]
s += " " + pad moves[i*board_width + j]
console.log s

BOARD_WIDTH = 8
MAX_NUM_SOLUTIONS = 100000

graph = knight_graph BOARD_WIDTH
tours = graph_tours graph, MAX_NUM_SOLUTIONS
console.log "#{tours.length} tours found (showing first and last)"
illustrate_knights_tour tours[0], BOARD_WIDTH
illustrate_knights_tour tours.pop(), BOARD_WIDTH

output

> time coffee knight.coffee
100000 tours found (showing first and last)

------
1 4 57 20 47 6 49 22
34 19 2 5 58 21 46 7
3 56 35 60 37 48 23 50
18 33 38 55 52 59 8 45
39 14 53 36 61 44 51 24
32 17 40 43 54 27 62 9
13 42 15 30 11 64 25 28
16 31 12 41 26 29 10 63

------
1 4 41 20 63 6 61 22
34 19 2 5 42 21 44 7
3 40 35 64 37 62 23 60
18 33 38 47 56 43 8 45
39 14 57 36 49 46 59 24
32 17 48 55 58 27 50 9
13 54 15 30 11 52 25 28
16 31 12 53 26 29 10 51

real 0m29.741s
user 0m25.656s
sys 0m0.253s

## D

### Fast Version

Translation of: C++
import std.stdio, std.algorithm, std.random, std.range,
std.conv, std.typecons, std.typetuple;

int[N][N] knightTour(size_t N=8)(in string start)
in {
assert(start.length >= 2);
} body {
static struct P { int x, y; }

immutable P[8] moves = [P(2,1), P(1,2), P(-1,2), P(-2,1),
P(-2,-1), P(-1,-2), P(1,-2), P(2,-1)];
int[N][N] data;

int[8] sortMoves(in int x, in int y) {
int[2][8] counts;
foreach (immutable i, immutable ref d1; moves) {
int c = 0;
foreach (immutable ref d2; moves) {
immutable p = P(x + d1.x + d2.x, y + d1.y + d2.y);
if (p.x >= 0 && p.x < N && p.y >= 0 && p.y < N &&
data[p.y][p.x] == 0)
c++;
}
counts[i] = [c, i];
}

counts[].randomShuffle; // Shuffle to randomly break ties.
counts[].sort(); // Lexicographic sort.

int[8] result = void;
transversal(counts[], 1).copy(result[]);
return result;
}

immutable p0 = P(start[0] - 'a', N - to!int(start[1 .. \$]));
data[p0.y][p0.x] = 1;

Tuple!(int, int, int, int[8])[N * N] order;
order[0] = tuple(p0.x, p0.y, 0, sortMoves(p0.x, p0.y));

int n = 0;
while (n < (N * N - 1)) {
immutable int x = order[n][0];
immutable int y = order[n][1];
bool ok = false;
foreach (immutable i; order[n][2] .. 8) {
immutable P d = moves[order[n][3][i]];
if (x+d.x < 0 || x+d.x >= N || y+d.y < 0 || y+d.y >= N)
continue;

if (data[y + d.y][x + d.x] == 0) {
order[n][2] = i + 1;
n++;
data[y + d.y][x + d.x] = n + 1;
order[n] = tuple(x+d.x,y+d.y,0,sortMoves(x+d.x,y+d.y));
ok = true;
break;
}
}

if (!ok) { // Failed. Backtrack.
data[y][x] = 0;
n--;
}
}

return data;
}

void main() {
foreach (immutable i, side; TypeTuple!(5, 8, 31, 101)) {
immutable form = "%(%" ~ text(side ^^ 2).length.text ~ "d %)";
foreach (ref row; ["c3", "b5", "a1", "a1"][i].knightTour!side)
writefln(form, row);
writeln();
}
}
Output:
23 16 11  6 21
10  5 22 17 12
15 24  1 20  7
4  9 18 13  2
25 14  3  8 19

63 20  3 24 59 36  5 26
2 23 64 37  4 25 58 35
19 62 21 50 55 60 27  6
22  1 54 61 38 45 34 57
53 18 49 44 51 56  7 28
12 15 52 39 46 31 42 33
17 48 13 10 43 40 29  8
14 11 16 47 30  9 32 41

275 112  19 116 277 604  21 118 823 770  23 120 961 940  25 122 943 926  27 124 917 898  29 126 911 872  31 128 197 870  33
18 115 276 601  20 117 772 767  22 119 958 851  24 121 954 941  26 123 936 925  28 125 912 899  30 127 910 871  32 129 198
111 274 113 278 605 760 603 822 771 824 769 948 957 960 939 944 953 942 927 916 929 918 897 908 913 900 873 196 875  34 869
114  17 600 273 602 775 766 773 768 949 850 959 852 947 952 955 932 937 930 935 924 915 920 905 894 909 882 901 868 199 130
271 110 279 606 759 610 761 776 821 764 825 816 951 956 853 938 945 934 923 928 919 896 893 914 907 904 867 874 195 876  35
16 581 272 599 280 607 774 765 762 779 950 849 826 815 946 933 854 931 844 857 890 921 906 895 886 883 902 881 200 131 194
109 270 281 580 609 758 611 744 777 820 763 780 817 848 827 808 811 846 855 922 843 858 889 892 903 866 885 192 877  36 201
282  15 582 269 598 579 608 757 688 745 778 819 754 783 814 847 828 807 810 845 856 891 842 859 884 887 880 863 202 193 132
267 108 283 578 583 612 689 614 743 756 691 746 781 818 753 784 809 812 829 806 801 840 835 888 865 862 203 878 191 530  37
14 569 268 585 284 597 576 619 690 687 742 755 692 747 782 813 752 785 802 793 830 805 860 841 836 879 864 529 204 133 190
107 266 285 570 577 584 613 686 615 620 695 684 741 732 711 748 739 794 751 786 803 800 839 834 861 528 837 188 531  38 205
286  13 568 265 586 575 596 591 618 685 616 655 696 693 740 733 712 749 738 795 792 831 804 799 838 833 722 527 206 189 134
263 106 287 508 571 590 587 574 621 592 639 694 683 656 731 710 715 734 787 750 737 796 791 832 721 798 207 532 187 474  39
12 417 264 567 288 509 572 595 588 617 654 657 640 697 680 713 730 709 716 735 788 727 720 797 790 723 526 473 208 135 186
105 262 289 416 507 566 589 512 573 622 593 638 653 682 659 698 679 714 729 708 717 736 789 726 719 472 533 184 475  40 209
290  11 418 261 502 415 510 565 594 513 562 641 658 637 652 681 660 699 678 669 728 707 718 675 724 525 704 471 210 185 136
259 104 291 414 419 506 503 514 511 564 623 548 561 642 551 636 651 670 661 700 677 674 725 706 703 534 211 476 183 396  41
10 331 260 493 292 501 420 495 504 515 498 563 624 549 560 643 662 635 650 671 668 701 676 673 524 705 470 395 212 137 182
103 258 293 330 413 494 505 500 455 496 547 516 485 552 625 550 559 644 663 634 649 672 667 702 535 394 477 180 397  42 213
294   9 332 257 492 329 456 421 490 499 458 497 546 517 484 553 626 543 558 645 664 633 648 523 666 469 536 393 220 181 138
255 102 295 328 333 412 491 438 457 454 489 440 459 486 545 518 483 554 627 542 557 646 665 632 537 478 221 398 179 214  43
8 319 256 335 296 345 326 409 422 439 436 453 488 441 460 451 544 519 482 555 628 541 522 647 468 631 392 219 222 139 178
101 254 297 320 327 334 411 346 437 408 423 368 435 452 487 442 461 450 445 520 481 556 629 538 479 466 399 176 215  44 165
298   7 318 253 336 325 344 349 410 347 360 407 424 383 434 427 446 443 462 449 540 521 480 467 630 391 218 223 164 177 140
251 100 303 300 321 316 337 324 343 350 369 382 367 406 425 384 433 428 447 444 463 430 539 390 465 400 175 216 169 166  45
6 299 252 317 304 301 322 315 348 361 342 359 370 381 366 405 426 385 432 429 448 389 464 401 174 217 224 163 150 141 168
99 250 241 302 235 248 307 338 323 314 351 362 341 358 371 380 365 404 377 386 431 402 173 388 225 160 153 170 167  46 143
240   5  98 249 242 305 234 247 308 339 232 313 352 363 230 357 372 379 228 403 376 387 226 159 154 171 162 149 142 151  82
63   2 239  66  97 236 243 306 233 246 309 340 231 312 353 364 229 356 373 378 227 158 375 172 161 148 155 152  83 144  47
4  67  64  61 238  69  96  59 244  71  94  57 310  73  92  55 354  75  90  53 374  77  88  51 156  79  86  49 146  81  84
1  62   3  68  65  60 237  70  95  58 245  72  93  56 311  74  91  54 355  76  89  52 157  78  87  50 147  80  85  48 145

### Shorter Version

import std.stdio, std.math, std.algorithm, std.range, std.typecons;

alias Square = Tuple!(int,"x", int,"y");

const(Square)[] knightTour(in Square[] board, in Square[] moves) pure @safe nothrow {
enum findMoves = (in Square sq) pure nothrow @safe =>
cartesianProduct([1, -1, 2, -2], [1, -1, 2, -2])
.filter!(ij => ij[0].abs != ij[1].abs)
.map!(ij => Square(sq.x + ij[0], sq.y + ij[1]))
.filter!(s => board.canFind(s) && !moves.canFind(s));
auto newMoves = findMoves(moves.back);
if (newMoves.empty)
return moves;
//alias warnsdorff = min!(s => findMoves(s).walkLength);
//immutable newSq = newMoves.dropOne.fold!warnsdorff(newMoves.front);
auto pairs = newMoves.map!(s => tuple(findMoves(s).walkLength, s));
immutable newSq = reduce!min(pairs.front, pairs.dropOne)[1];
return board.knightTour(moves ~ newSq);
}

void main(in string[] args) {
enum toSq = (in string xy) => Square(xy[0] - '`', xy[1] - '0');
immutable toAlg = (in Square s) => [dchar(s.x + '`'), dchar(s.y + '0')];
immutable sq = toSq((args.length == 2) ? args[1] : "e5");
const board = iota(1, 9).cartesianProduct(iota(1, 9)).map!Square.array;
writefln("%(%-(%s -> %)\n%)", board.knightTour([sq]).map!toAlg.chunks(8));
}
Output:
e5 -> d7 -> b8 -> a6 -> b4 -> a2 -> c1 -> b3
a1 -> c2 -> a3 -> b1 -> d2 -> f1 -> h2 -> g4
h6 -> g8 -> e7 -> c8 -> a7 -> c6 -> a5 -> b7
d8 -> f7 -> h8 -> g6 -> f8 -> h7 -> f6 -> e8
g7 -> h5 -> g3 -> h1 -> f2 -> d1 -> b2 -> a4
b6 -> a8 -> c7 -> b5 -> c3 -> d5 -> e3 -> c4
d6 -> e4 -> c5 -> d3 -> e1 -> g2 -> h4 -> f5
d4 -> e2 -> f4 -> e6 -> g5 -> f3 -> g1 -> h3

## EchoLisp

The algorithm uses iterative backtracking and Warnsdorff's heuristic. It can output closed or non-closed tours.

(require 'plot)
(define *knight-moves*
'((2 . 1)(2 . -1 ) (1 . -2) (-1 . -2 )(-2 . -1) (-2 . 1) (-1 . 2) (1 . 2)))
(define *hit-squares* null)
(define *legal-moves* null)
(define *tries* 0)

(define (square x y n ) (+ y (* x n)))
(define (dim n) (1- (* n n))) ; n^2 - 1

;; check legal knight move from sq
;; return null or (list destination-square)

(define (legal-disp n sq k-move)
(let ((x (+ (quotient sq n) (first k-move)))
(y (+ (modulo sq n) (rest k-move))))
(if (and (>= x 0) (< x n) (>= y 0) (< y n))
(list (square x y n)) null)))

;; list of legal destination squares from sq
(define (legal-moves sq k-moves n )
(if (null? k-moves) null
(append (legal-moves sq (rest k-moves) n) (legal-disp n sq (first k-moves)))))

;; square freedom = number of destination squares not already reached
(define (freedom sq)
(for/sum ((dest (vector-ref *legal-moves* sq)))
(if (vector-ref *hit-squares* dest) 0 1)))

;; The chess adage" A knight on the rim is dim" is false here :
;; choose to move to square with smallest freedom : Warnsdorf's rule
(define (square-sort a b)
(< (freedom a) (freedom b)))

;; knight tour engine
(define (play sq step starter last-one wants-open)
(set! *tries* (1+ *tries*))
(vector-set! *hit-squares* sq step) ;; flag used square
(if (= step last-one) (throw 'HIT last-one)) ;; stop on first path found

(when (or wants-open ;; cut search iff closed path
(and (< step last-one) (> (freedom starter) 0))) ;; this ensures a closed path

(for ((target (list-sort square-sort (vector-ref *legal-moves* sq))))
(unless (vector-ref *hit-squares* target)
(play target (1+ step) starter last-one wants-open))))
(vector-set! *hit-squares* sq #f)) ;; unflag used square

(define (show-steps n wants-open)
(string-delimiter "")
(if wants-open
(printf "♘-tour: %d tries." *tries*)
(printf "♞-closed-tour: %d tries." *tries*))
(for ((x n))
(writeln)
(for((y n))
(write (string-pad-right (vector-ref *hit-squares* (square x y n)) 4)))))

(define (k-tour (n 8) (starter 0) (wants-open #t))
(set! *hit-squares* (make-vector (* n n) #f))
;; build vector of legal moves for squares 0..n^2-1
(set! *legal-moves*
(build-vector (* n n) (lambda(sq) (legal-moves sq *knight-moves* n))))
(set! *tries* 0) ; counter
(try
(play starter 0 starter (dim n) wants-open)
(catch (hit mess) (show-steps n wants-open))))

Output:

(k-tour 8 0 #f)
♞-closed-tour: 66 tries.
0 47 14 31 62 27 12 29
15 32 63 54 13 30 57 26
48 1 46 61 56 59 28 11
33 16 55 50 53 44 25 58
2 49 42 45 60 51 10 39
17 34 19 52 43 40 7 24
20 3 36 41 22 5 38 9
35 18 21 4 37 8 23 6

(k-tour 20 57)
♘-tour: 400 tries.
31 34 29 104 209 36 215 300 211 38 213 354 343 40 345 386 383 42 1 388
28 103 32 35 216 299 210 37 214 335 342 39 346 385 382 41 390 387 396 43
33 30 105 208 201 308 301 336 323 212 353 340 355 344 391 384 395 0 389 2
102 27 202 219 298 217 322 309 334 341 356 347 358 351 376 381 378 399 44 397
203 106 207 200 307 228 311 302 337 324 339 352 373 364 379 392 375 394 3 368
26 101 220 229 218 297 304 321 310 333 348 357 350 359 374 377 380 367 398 45
107 204 199 206 227 306 231 312 303 338 325 330 363 372 365 328 393 254 369 4
100 25 122 221 230 233 296 305 320 313 332 349 326 329 360 371 366 251 46 253
121 108 205 198 145 226 237 232 295 286 319 314 331 362 327 316 255 370 5 178
24 99 144 123 222 129 234 279 236 281 294 289 318 315 256 361 250 179 252 47
109 120 111 130 197 146 225 238 285 278 287 272 293 290 317 180 257 162 177 6
98 23 124 143 128 223 276 235 280 239 282 291 288 265 270 249 176 181 48 161
115 110 119 112 131 196 147 224 277 284 273 266 271 292 245 258 163 174 7 58
22 97 114 125 142 127 140 275 194 267 240 283 264 269 248 175 182 59 160 49
87 116 95 118 113 132 195 148 187 274 263 268 191 244 259 246 173 164 57 8
96 21 88 133 126 141 150 139 262 193 190 241 260 247 172 183 60 159 50 65
77 86 117 94 89 138 135 188 149 186 261 192 171 184 243 156 165 64 9 56
20 81 78 85 134 93 90 151 136 189 170 185 242 155 166 61 158 53 66 51
79 76 83 18 91 74 137 16 169 72 153 14 167 70 157 12 63 68 55 10
82 19 80 75 84 17 92 73 152 15 168 71 154 13 62 69 54 11 52 67

Plotting

64 shades of gray. We plot the move sequence in shades of gray, from black to white. The starting square is red. The ending square is green. One can observe that the squares near the border are played first (dark squares).

(define (step-color x y n last-one)
(letrec ((sq (square (floor x) (floor y) n))
(step (vector-ref *hit-squares* sq) n n))
(cond ((= 0 step) (rgb 1 0 0)) ;; red starter
((= last-one step) (rgb 0 1 0)) ;; green end
(else (gray (// step n n))))))

(define ( k-plot n)
(plot-rgb (lambda (x y) (step-color x y n (dim n))) (- n epsilon) (- n epsilon)))

Closed path on a 12x12 board: [1]

Open path on a 24x24 board: [2]

## Elixir

Translation of: Ruby
defmodule Board do
import Integer, only: [is_odd: 1]

defmodule Cell do
end

defp initialize(rows, cols) do
board = for i <- 1..rows, j <- 1..cols, into: %{}, do: {{i,j}, true}
for i <- 1..rows, j <- 1..cols, into: %{} do
end
end

defp solve(board, ij, num, goal) do
board = Map.update!(board, ij, fn cell -> %{cell | value: num} end)
if num == goal do
throw({:ok, board})
else
wdof(board, ij)
|> Enum.each(fn k -> solve(board, k, num+1, goal) end)
end
end

defp wdof(board, ij) do # Warnsdorf's rule
|> Enum.filter(fn k -> board[k].value == 0 end)
|> Enum.sort_by(fn k ->
Enum.count(board[k].adj, fn x -> board[x].value == 0 end)
end)
end

defp to_string(board, rows, cols) do
width = to_string(rows * cols) |> String.length
format = String.duplicate("~#{width}w ", cols)
Enum.map_join(1..rows, "\n", fn i ->
:io_lib.fwrite format, (for j <- 1..cols, do: board[{i,j}].value)
end)
end

def knight_tour(rows, cols, sx, sy) do
IO.puts "\nBoard (#{rows} x #{cols}), Start: [#{sx}, #{sy}]"
if is_odd(rows*cols) and is_odd(sx+sy) do
IO.puts "No solution"
else
try do
initialize(rows, cols)
|> solve({sx,sy}, 1, rows*cols)
IO.puts "No solution"
catch
{:ok, board} -> IO.puts to_string(board, rows, cols)
end
end
end
end

Board.knight_tour(8,8,4,2)
Board.knight_tour(5,5,3,3)
Board.knight_tour(4,9,1,1)
Board.knight_tour(5,5,1,2)
Board.knight_tour(12,12,2,2)
Output:
Board (8 x 8), Start: [4, 2]
23 20  3 32 25 10  5  8
2 35 24 21  4  7 26 11
19 22 33 36 31 28  9  6
34  1 50 29 48 37 12 27
51 18 53 44 61 30 47 38
54 43 56 49 58 45 62 13
17 52 41 60 15 64 39 46
42 55 16 57 40 59 14 63

Board (5 x 5), Start: [3, 3]
19  8  3 14 25
2 13 18  9  4
7 20  1 24 15
12 17 22  5 10
21  6 11 16 23

Board (4 x 9), Start: [1, 1]
1 34  3 28 13 24  9 20 17
4 29  6 33  8 27 18 23 10
35  2 31 14 25 12 21 16 19
30  5 36  7 32 15 26 11 22

Board (5 x 5), Start: [1, 2]
No solution

Board (12 x 12), Start: [2, 2]
87  24  59   2  89  26  61   4  39   8  31   6
58   1  88  25  60   3  92  27  30   5  38   9
23  86  83  90 103  98  29  62  93  40   7  32
82  57 102  99  84  91 104  97  28  37  10  41
101  22  85 114 105 116 111  94  63  96  33  36
56  81 100 123 128 113 106 117 110  35  42  11
21 122 141  80 115 124 127 112  95  64 109  34
144  55  78 121 142 129 118 107 126 133  12  43
51  20 143 140  79 120 125 138  69 108  65 134
54  73  52  77 130 139  70 119 132 137  44  13
19  50  75  72  17  48 131  68  15  46 135  66
74  53  18  49  76  71  16  47 136  67  14  45

## Elm

module Main exposing (main)

import Browser exposing (element)
import Html as H
import Html.Attributes as HA
import List exposing (filter, head, length, map, map2, member, tail)
import List.Extra exposing (andThen, minimumBy)
import String exposing (join)
import Svg exposing (g, line, rect, svg)
import Svg.Attributes exposing (fill, height, style, version, viewBox, width, x, x1, x2, y, y1, y2)
import Svg.Events exposing (onClick)
import Time exposing (every)
import Tuple

type alias Cell =
( Int, Int )

type alias BoardSize =
( Int, Int )

type alias Model =
{ path : List Cell
, board : List Cell
, pause_ms : Float
, size : BoardSize
}

type Msg
= Tick Time.Posix
| SetStart Cell
| SetSize BoardSize
| SetPause Float

boardsize_width: BoardSize -> Int
boardsize_width bs =
Tuple.second bs

boardsize_height: BoardSize -> Int
boardsize_height bs =
Tuple.first bs

boardsize_dec: Int -> Int
boardsize_dec n =
let
minimum_size = 3
in
if n <= minimum_size then
minimum_size
else
n - 1
boardsize_inc: Int -> Int
boardsize_inc n =
let
maximum_size = 40
in
if n >= maximum_size then
maximum_size
else
n + 1

pause_inc: Float -> Float
pause_inc n =
n + 10

-- decreasing pause time (ms) increases speed
pause_dec: Float -> Float
pause_dec n =
let
minimum_pause = 0
in
if n <= minimum_pause then
minimum_pause
else
n - 10

board_init : BoardSize -> List Cell
board_init board_size =
List.range 0 (boardsize_height board_size - 1)
|> andThen
(\r ->
List.range 0 (boardsize_width board_size - 1)
|> andThen
(\c ->
[ ( r, c ) ]
)
)

nextMoves : Model -> Cell -> List Cell
nextMoves model ( stRow, stCol ) =
let
c =
[ 1, 2, -1, -2 ]

km =
c
|> andThen
(\cRow ->
c
|> andThen
(\cCol ->
if abs cRow == abs cCol then
[]

else
[ ( cRow, cCol ) ]
)
)

jumps =
List.map (\( kmRow, kmCol ) -> ( kmRow + stRow, kmCol + stCol )) km
in
List.filter (\j -> List.member j model.board && not (List.member j model.path)) jumps

bestMove : Model -> Maybe Cell
bestMove model =
Just mph ->
minimumBy (List.length << nextMoves model) (nextMoves model mph)
_ ->
Nothing

-- Initialize the application - https://guide.elm-lang.org/effects/
init : () -> ( Model, Cmd Msg )
init _ =
let
-- Initial board height and width
initial_size =
8

-- Initial chess board
initial_board =
board_init (initial_size, initial_size)

initial_path =
[]
initial_pause =
10
in
( Model initial_path initial_board initial_pause (initial_size, initial_size), Cmd.none )

-- View the model - https://guide.elm-lang.org/effects/
view : Model -> H.Html Msg
view model =
let
showChecker row col =
rect
[ x <| String.fromInt col
, y <| String.fromInt row
, width "1"
, height "1"
, fill <|
if modBy 2 (row + col) == 0 then
"blue"

else
"grey"
, onClick <| SetStart ( row, col )
]
[]

showMove ( row0, col0 ) ( row1, col1 ) =
line
[ x1 <| String.fromFloat (toFloat col0 + 0.5)
, y1 <| String.fromFloat (toFloat row0 + 0.5)
, x2 <| String.fromFloat (toFloat col1 + 0.5)
, y2 <| String.fromFloat (toFloat row1 + 0.5)
, style "stroke:yellow;stroke-width:0.05"
]
[]

render mdl =
let
checkers =
mdl.board
|> andThen
(\( r, c ) ->
[ showChecker r c ]
)

moves =
case List.tail mdl.path of
Nothing ->
[]

Just tl ->
List.map2 showMove mdl.path tl
in
checkers ++ moves

unvisited =
length model.board - length model.path

center =
[ HA.style "text-align" "center" ]

table =
[ HA.style "text-align" "center", HA.style "display" "table", HA.style "width" "auto", HA.style "margin" "auto" ]
table_row =
[ HA.style "display" "table-row", HA.style "width" "auto" ]

table_cell =
[ HA.style "display" "table-cell", HA.style "width" "auto", HA.style "padding" "1px 3px" ]
rows =
boardsize_height model.size

cols =
boardsize_width model.size
in
H.div
[]
[ H.h1 center [ H.text "Knight's Tour" ]
-- controls
, H.div
table
[ H.div -- labels
table_row
[ H.div
table_cell
[ H.text "Rows"]
, H.div
table_cell
[ H.text "Columns"]
, H.div
table_cell
[ H.text ""]
, H.div
table_cell
[ H.text "Pause (ms)"]
]
, H.div
table_row
[ H.div -- Increase
table_cell
[ H.button [onClick <| SetSize ( boardsize_inc rows, cols )] [ H.text "▲"] ]
, H.div
table_cell
[ H.button [onClick <| SetSize ( rows, boardsize_inc cols )] [ H.text "▲"] ]
, H.div
table_cell
[ H.text ""]
, H.div
table_cell
[ H.button [onClick <| SetPause ( pause_inc model.pause_ms )] [ H.text "▲"] ]
]
, H.div
table_row
[ H.div -- Value
table_cell
[ H.text <| String.fromInt rows ]
, H.div
table_cell
[ H.text <| String.fromInt cols]
, H.div
table_cell
[ H.text ""]
, H.div
table_cell
[ H.text <| String.fromFloat model.pause_ms]
]
, H.div
table_row
[ H.div -- Decrease
table_cell
[ H.button [onClick <| SetSize ( boardsize_dec rows, cols )] [ H.text "▼"] ]
, H.div
table_cell
[ H.button [onClick <| SetSize ( rows, boardsize_dec cols )] [ H.text "▼"] ]
, H.div
table_cell
[ H.text ""]
, H.div
table_cell
[ H.button [onClick <| SetPause ( pause_dec model.pause_ms )] [ H.text "▼"] ]
]
]
, H.h2 center [ H.text "(pick a square)" ]
, H.div -- chess board
center
[ svg
[ version "1.1"
, width (String.fromInt (25 * cols))
, height (String.fromInt (25 * rows))
, viewBox
(join " "
[ String.fromInt 0
, String.fromInt 0
, String.fromInt cols
, String.fromInt rows
]
)
]
[ g [] <| render model ]
]
, H.h3 center [ H.text <| "Unvisited count : " ++ String.fromInt unvisited ]
]

-- Update the model - https://guide.elm-lang.org/effects/
update : Msg -> Model -> ( Model, Cmd Msg )
update msg model =
let
mo =
case msg of
SetPause pause ->
{ model | pause_ms = pause }

SetSize board_size ->
{ model | board = board_init board_size, path = [], size = board_size }

SetStart start ->
{ model | path = [ start ] }

Tick _ ->
case model.path of
[] ->
model

_ ->
case bestMove model of
Nothing ->
model

Just best ->
{ model | path = best :: model.path }
in
( mo, Cmd.none )

-- Subscribe to https://guide.elm-lang.org/effects/
subscriptions : Model -> Sub Msg
subscriptions model =
Time.every model.pause_ms Tick

-- Application entry point
main: Program () Model Msg
main =
element -- https://package.elm-lang.org/packages/elm/browser/latest/Browser#element
{ init = init
, view = view
, update = update
, subscriptions = subscriptions
}

## Erlang

Again I use backtracking. It seemed easier this time.

-module( knights_tour ).

display( Moves ) ->
%% The knigh walks the moves {Position, Step_nr} order.
%% Top left corner is {\$a, 8}, Bottom right is {\$h, 1}.
io:fwrite( "Moves:" ),
lists:foldl( fun display_moves/2, erlang:length(Moves), lists:keysort(2, Moves) ),
io:nl(),
[display_row(Y, Moves) || Y <- lists:seq(8, 1, -1)].

solve( First_square ) ->
try
bt_loop( 1, next_moves(First_square), [{First_square, 1}] )

catch
_:{ok, Moves} -> Moves

end.

io:fwrite( "Starting {a, 1}~n" ),
Moves = solve( {\$a, 1} ),
display( Moves ).

bt( N, Move, Moves ) -> bt_reject( is_not_allowed_knight_move(Move, Moves), N, Move, [{Move, N} | Moves] ).

bt_accept( true, _N, _Move, Moves ) -> erlang:throw( {ok, Moves} );
bt_accept( false, N, Move, Moves ) -> bt_loop( N, next_moves(Move), Moves ).

bt_loop( N, New_moves, Moves ) -> [bt( N+1, X, Moves ) || X <- New_moves].

bt_reject( true, _N, _Move, _Moves ) -> backtrack;
bt_reject( false, N, Move, Moves ) -> bt_accept( is_all_knights(Moves), N, Move, Moves ).

display_moves( {{X, Y}, 1}, Max ) ->
io:fwrite(" ~p. N~c~p", [1, X, Y]),
Max;
display_moves( {{X, Y}, Max}, Max ) ->
io:fwrite(" N~c~p~n", [X, Y]),
Max;
display_moves( {{X, Y}, Step_nr}, Max ) when Step_nr rem 8 =:= 0 ->
io:fwrite(" N~c~p~n~p. N~c~p", [X, Y, Step_nr, X, Y]),
Max;
display_moves( {{X, Y}, Step_nr}, Max ) ->
io:fwrite(" N~c~p ~p. N~c~p", [X, Y, Step_nr, X, Y]),
Max.

display_row( Row, Moves ) ->
[io:fwrite(" ~2b", [proplists:get_value({X, Row}, Moves)]) || X <- [\$a, \$b, \$c, \$d, \$e, \$f, \$g, \$h]],
io:nl().

is_all_knights( Moves ) when erlang:length(Moves) =:= 64 -> true;
is_all_knights( _Moves ) -> false.

is_asymetric( Start_column, Start_row, Stop_column, Stop_row ) ->
erlang:abs( Start_column - Stop_column ) =/= erlang:abs( Start_row - Stop_row ).

is_not_allowed_knight_move( Move, Moves ) ->
no_such_move =/= proplists:get_value( Move, Moves, no_such_move ).

next_moves( {Column, Row} ) ->
[{X, Y} || X <- next_moves_column(Column), Y <- next_moves_row(Row), is_asymetric(Column, Row, X, Y)].

next_moves_column( \$a ) -> [\$b, \$c];
next_moves_column( \$b ) -> [\$a, \$c, \$d];
next_moves_column( \$g ) -> [\$e, \$f, \$h];
next_moves_column( \$h ) -> [\$g, \$f];
next_moves_column( C ) -> [C - 2, C - 1, C + 1, C + 2].

next_moves_row( 1 ) -> [2, 3];
next_moves_row( 2 ) -> [1, 3, 4];
next_moves_row( 7 ) -> [5, 6, 8];
next_moves_row( 8 ) -> [6, 7];
next_moves_row( N ) -> [N - 2, N - 1, N + 1, N + 2].

Output:
Starting {a, 1}
Moves: 1. Na1 Nb3 2. Nb3 Na5 3. Na5 Nb7 4. Nb7 Nc5 5. Nc5 Na4 6. Na4 Nb2 7. Nb2 Nc4
8. Nc4 Na3 9. Na3 Nb1 10. Nb1 Nc3 11. Nc3 Na2 12. Na2 Nb4 13. Nb4 Na6 14. Na6 Nb8 15. Nb8 Nc6
16. Nc6 Na7 17. Na7 Nb5 18. Nb5 Nc7 19. Nc7 Na8 20. Na8 Nb6 21. Nb6 Nc8 22. Nc8 Nd6 23. Nd6 Ne4
24. Ne4 Nd2 25. Nd2 Nf1 26. Nf1 Ne3 27. Ne3 Nc2 28. Nc2 Nd4 29. Nd4 Ne2 30. Ne2 Nc1 31. Nc1 Nd3
32. Nd3 Ne1 33. Ne1 Ng2 34. Ng2 Nf4 35. Nf4 Nd5 36. Nd5 Ne7 37. Ne7 Ng8 38. Ng8 Nh6 39. Nh6 Nf5
40. Nf5 Nh4 41. Nh4 Ng6 42. Ng6 Nh8 43. Nh8 Nf7 44. Nf7 Nd8 45. Nd8 Ne6 46. Ne6 Nf8 47. Nf8 Nd7
48. Nd7 Ne5 49. Ne5 Ng4 50. Ng4 Nh2 51. Nh2 Nf3 52. Nf3 Ng1 53. Ng1 Nh3 54. Nh3 Ng5 55. Ng5 Nh7
56. Nh7 Nf6 57. Nf6 Ne8 58. Ne8 Ng7 59. Ng7 Nh5 60. Nh5 Ng3 61. Ng3 Nh1 62. Nh1 Nf2 63. Nf2 Nd1

20 15 22 45 58 47 38 43
17  4 19 48 37 44 59 56
14 21 16 23 46 57 42 39
3 18  5 36 49 40 55 60
6 13  8 29 24 35 50 41
9  2 11 32 27 52 61 54
12  7 28 25 30 63 34 51
1 10 31 64 33 26 53 62

## ERRE

Taken from ERRE distribution disk. Comments are in Italian.

! **********************************************************************
! * *
! * IL GIRO DEL CAVALLO - come collocare un cavallo su di una *
! * scacchiera n*n passando una sola volta *
! * per ogni casella. *
! * *
! **********************************************************************
! ----------------------------------------------------------------------
! Inizializzazione dei parametri
! ----------------------------------------------------------------------

PROGRAM KNIGHT

!\$INTEGER
!\$KEY

DIM H[25,25],A[8],B[8],P0[8],P1[8]

!\$INCLUDE="PC.LIB"

PROCEDURE INIT_SCACCHIERA
! **********************************************************************
! * Routine di inizializzazione scacchiera *
! **********************************************************************
FOR I1=1 TO 8 DO
U=X+A[I1] V=Y+B[I1]
IF (U>0 AND U<=N) AND (V>0 AND V<=N) THEN
H[U,V]=H[U,V]-1
END IF
END FOR
END PROCEDURE

PROCEDURE MOSTRA_SCACCHIERA
! *********************************************************************
! * Routine di visualizzazione della scacchiera *
! *********************************************************************
LOCATE(5,1) COLOR(0,7) PRINT(" Mossa num.";NMOS) COLOR(7,0)
L2=N
FOR I2=1 TO N DO
PRINT
FOR L1=1 TO N DO
IF H[L1,L2]>0 THEN COLOR(15,0) END IF
WRITE("####";H[L1,L2];)
COLOR(7,0)
END FOR
L2=L2-1
END FOR
END PROCEDURE

PROCEDURE AGGIORNA_SCACCHIERA
! *********************************************************************
! * Routine di Aggiornamento Scacchiera *
! *********************************************************************
B=1
FOR I1=1 TO 8 DO
U=X+A[I1] V=Y+B[I1]
IF (U>0 AND U<=N) AND (V>0 AND V<=N) THEN
IF H[U,V]<=0 THEN
H[U,V]=H[U,V]+1 B=0
END IF
END IF
END FOR
IF B=1 THEN Q1=0 END IF
END PROCEDURE

PROCEDURE MOSSA_MAX_PESO
! *********************************************************************
! * Cerca la prossima mossa con il massimo peso *
! *********************************************************************
M1=0 RO=1
FOR W=1 TO 8 DO
U=Z1+A[W] V=Z2+B[W]
IF (U>0 AND U<=N) AND (V>0 AND V<=N) THEN
IF H[U,V]<=0 AND H[U,V]<=M1 THEN
IF H[U,V]=M1 THEN
RO=RO+1 P0[RO]=W
ELSE
M1=H[U,V] Q1=1 T1=U T2=V RO=1 P0[1]=W
END IF
END IF
END IF
END FOR
END PROCEDURE

PROCEDURE MOSSA_MIN_PESO
! *********************************************************************
! * Cerca la prossima mossa con il minimo peso *
! *********************************************************************
M1=-9 RO=1
FOR W=1 TO 8 DO
U=Z1+A[W] V=Z2+B[W]
IF (U>0 AND U<=N) AND (V>0 AND V<=N) THEN
IF H[U,V]<=0 AND H[U,V]>=M1 THEN
IF H[U,V]=M1 THEN
RO=RO+1 P0[RO]=W
ELSE
M1=H[U,V] Q1=1 T1=U T2=V RO=1 P0[1]=W
END IF
END IF
END IF
END FOR
END PROCEDURE

BEGIN
A[1]=1 A[2]=2 A[3]=2 A[4]=1
A[5]=-1 A[6]=-2 A[7]=-2 A[8]=-1
B[1]=2 B[2]=1 B[3]=-1 B[4]=-2
B[5]=-2 B[6]=-1 B[7]=1 B[8]=2

CLS
PRINT(" *** LA GALOPPATA DEL CAVALIERE ***")
PRINT
PRINT("Inserire la dimensione della scacchiera (max. 25)";)
INPUT(N)
PRINT("Inserire la caselle di partenza (x,y) ";)
INPUT(X1,Y1)
NMOS=1 A1=1 N1=N*N ESCAPE=FALSE
! ----------------------------------------------------------------------
! Set della scacchiera
! ----------------------------------------------------------------------
WHILE NOT ESCAPE DO
FOR I=1 TO N DO
FOR J=1 TO N DO
H[I,J]=0
END FOR
END FOR
FOR I=1 TO N DO
FOR J=1 TO N DO
X=I Y=J
INIT_SCACCHIERA
END FOR
END FOR

! ----------------------------------------------------------------------
! Effettua la prima mossa
! ----------------------------------------------------------------------
X=X1 Y=Y1 H[X,Y]=1 L=2
AGGIORNA_SCACCHIERA
Q1=1 Q2=1
! -----------------------------------------------------------------------
! Trova la prossima mossa
! -----------------------------------------------------------------------
WHILE Q1<>0 AND Q2<>0 DO
Q1=0 Z1=X Z2=Y
MOSSA_MIN_PESO
IF RO<=1 THEN
C1=T1 C2=T2
ELSE
! ------------------------------------------------------------------------
! Esamina tutti i vincoli
! ------------------------------------------------------------------------
FOR K=1 TO RO DO
P1[K]=P0[K]
END FOR
R1=RO
IF A1=1 THEN M2=-9 ELSE M2=0 END IF
FOR K=1 TO R1 DO
F1=P1[K] Z1=X+A[F1] Z2=Y+B[F1]
IF A1=1 THEN
MOSSA_MAX_PESO
IF M1<=M2 THEN
!\$NULL
ELSE
M2=M1 C1=Z1 C2=Z2
END IF
ELSE
MOSSA_MIN_PESO
IF M1>=M2 THEN
!\$NULL
ELSE
M2=M1 C1=Z1 C2=Z2
END IF
END IF
END FOR
! ------------------------------------------------------------------------
! Prossima mossa trovata:aggiorna la scacchiera
! ------------------------------------------------------------------------
END IF
IF Q1<>0 THEN
X=C1 Y=C2 H[X,Y]=L
AGGIORNA_SCACCHIERA
IF L=N1 THEN Q2=0 END IF
END IF
L=L+1
MOSTRA_SCACCHIERA
NMOS=NMOS+1
END WHILE
! ------------------------------------------------------------------------
! La ricerca è terminata: visualizza i risultati
! ------------------------------------------------------------------------
PRINT PRINT
IF Q2<>1 THEN
PRINT("*** Trovata la soluzione! ***")
MOSTRA_SCACCHIERA
ESCAPE=TRUE
ELSE
IF A1=0 THEN
PRINT("Nessuna soluzione.")
ESCAPE=TRUE
ELSE
BEEP
A1=0
END IF
END IF
END WHILE
REPEAT
GET(A\$)
UNTIL A\$<>""
END PROGRAM

Output:
***    LA GALOPPATA DEL CAVALIERE    ***

Inserire la dimensione della scacchiera (max. 25)? 8
Inserire la caselle di partenza (x,y) ? 1,1
Mossa num. 64

64   7  54  41  60   9  48  39
53  42  61   8  55  40  35  10
6  63  44  59  34  49  38  47
43  52  21  62  45  56  11  36
20   5  58  33  50  37  46  25
31   2  51  22  57  26  15  12
4  19  32  29  14  17  24  27
1  30   3  18  23  28  13  16

*** Trovata la soluzione! ***

## FreeBASIC

Dim Shared As Integer tamano, xc, yc, nm
Dim As Integer f, qm, nmov, n = 0
Dim As String posini

Cls : Color 11
Input "Tamaño tablero: ", tamano
Input "Posicion inicial: ", posini

Dim As Integer x = Asc(Mid(posini,1,1))-96
Dim As Integer y = Val(Mid(posini,2,1))
Dim Shared As Integer tablero(tamano,tamano), dx(8), dy(8)
For f = 1 To 8 : Read dx(f), dy(f) : Next f
Data 2,1,1,2,-1,2,-2,1,-2,-1,-1,-2,1,-2,2,-1

Sub FindMoves()
Dim As Integer i, xt, yt
If xc < 1 Or yc < 1 Or xc > tamano Or yc > tamano Then nm = 1000: Return
If tablero(xc,yc) Then nm = 2000: Return
nm = 0
For i = 1 To 8
xt = xc+dx(i)
yt = yc+dy(i)
If xt < 1 Or yt < 1 Or xt > tamano Or yt > tamano Then 'Salta este movimiento
Elseif tablero(xt,yt) Then 'Salta este movimiento
Else
nm += 1
End If
Next i
End Sub

Color 4, 7 'Pinta tablero
For f = 1 To tamano
Locate 15-tamano, 3*f: Print " "; Chr(96+f); " ";
Locate 17-f, 3*(tamano+1)+1: Print Using "##"; f;
Next f

Color 15, 0
Do
n += 1
tablero(x,y) = n
Locate 17-y, 3*x: Print Using "###"; n;
If n = tamano*tamano Then Exit Do
nmov = 100
For f = 1 To 8
xc = x+dx(f)
yc = y+dy(f)
FindMoves()
If nm < nmov Then nmov = nm: qm = f
Next f
x = x+dx(qm)
y = y+dy(qm)
Sleep 1
Loop
Color 14 : Locate Csrlin+tamano, 1
Print " Pulsa cualquier tecla para finalizar..."
Sleep
End

Output:
Tamaño tablero:  8
Posicion inicial: c3

a  b  c  d  e  f  g  h

24 11 22 19 26  9 38 47  8
21 18 25 10 39 48 27  8  7
12 23 20 53 28 37 46 49  6
17 52 29 40 59 50  7 36  5
30 13 58 51 54 41 62 45  4
57 16  1 42 63 60 35  6  3
2 31 14 55  4 33 44 61  2
15 56  3 32 43 64  5 34  1

Pulsa cualquier tecla para finalizar...

## Fōrmulæ

Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for storage and transfer purposes more than visualization and edition.

Programs in Fōrmulæ are created/edited online in its website, However they run on execution servers. By default remote servers are used, but they are limited in memory and processing power, since they are intended for demonstration and casual use. A local server can be downloaded and installed, it has no limitations (it runs in your own computer). Because of that, example programs can be fully visualized and edited, but some of them will not run if they require a moderate or heavy computation/memory resources, and no local server is being used.

## Go

### Warnsdorf's rule

package main

import (
"fmt"
"math/rand"
"time"
)

// input, 0-based start position
const startRow = 0
const startCol = 0

func main() {
rand.Seed(time.Now().Unix())
for !knightTour() {
}
}

var moves = []struct{ dr, dc int }{
{2, 1},
{2, -1},
{1, 2},
{1, -2},
{-1, 2},
{-1, -2},
{-2, 1},
{-2, -1},
}

// Attempt knight tour starting at startRow, startCol using Warnsdorff's rule
// and random tie breaking. If a tour is found, print it and return true.
// Otherwise no backtracking, just return false.
func knightTour() bool {
// 8x8 board. squares hold 1-based visit order. 0 means unvisited.
board := make([][]int, 8)
for i := range board {
board[i] = make([]int, 8)
}
r := startRow
c := startCol
board[r][c] = 1 // first move
for move := 2; move <= 64; move++ {
minNext := 8
var mr, mc, nm int
candidateMoves:
for _, cm := range moves {
cr := r + cm.dr
if cr < 0 || cr >= 8 { // off board
continue
}
cc := c + cm.dc
if cc < 0 || cc >= 8 { // off board
continue
}
if board[cr][cc] > 0 { // already visited
continue
}
// cr, cc candidate legal move.
p := 0 // count possible next moves.
for _, m2 := range moves {
r2 := cr + m2.dr
if r2 < 0 || r2 >= 8 {
continue
}
c2 := cc + m2.dc
if c2 < 0 || c2 >= 8 {
continue
}
if board[r2][c2] > 0 {
continue
}
p++
if p > minNext { // bail out as soon as it's eliminated
continue candidateMoves
}
}
if p < minNext { // it's better. keep it.
minNext = p // new min possible next moves
nm = 1 // number of candidates with this p
mr = cr // best candidate move
mc = cc
continue
}
// it ties for best so far.
// keep it with probability 1/(number of tying moves)
nm++ // number of tying moves
if rand.Intn(nm) == 0 { // one chance to keep it
mr = cr
mc = cc
}
}
if nm == 0 { // no legal move
return false
}
// make selected move
r = mr
c = mc
board[r][c] = move
}
// tour complete. print board.
for _, r := range board {
for _, m := range r {
fmt.Printf("%3d", m)
}
fmt.Println()
}
return true
}
Output:
1  4 39 20 23  6 63 58
40 19  2  5 62 57 22  7
3 38 41 48 21 24 59 64
18 43 32 37 56 61  8 25
31 14 47 42 49 36 53 60
46 17 44 33 52 55 26  9
13 30 15 50 11 28 35 54
16 45 12 29 34 51 10 27

### Ant colony

/* Adapted from "Enumerating Knight's Tours using an Ant Colony Algorithm"
by Philip Hingston and Graham Kendal,
PDF at http://www.cs.nott.ac.uk/~gxk/papers/cec05knights.pdf. */

package main

import (
"fmt"
"math/rand"
"sync"
"time"
)

const boardSize = 8
const nSquares = boardSize * boardSize
const completeTour = nSquares - 1

// task input: starting square. These are 1 based, but otherwise 0 based
// row and column numbers are used througout the program.
const rStart = 2
const cStart = 3

// pheromone representation read by ants
var tNet = make([]float64, nSquares*8)

// row, col deltas of legal moves
var drc = [][]int{{1, 2}, {2, 1}, {2, -1}, {1, -2},
{-1, -2}, {-2, -1}, {-2, 1}, {-1, 2}}

// get square reached by following edge k from square (r, c)
func dest(r, c, k int) (int, int, bool) {
r += drc[k][0]
c += drc[k][1]
return r, c, r >= 0 && r < boardSize && c >= 0 && c < boardSize
}

// struct represents a pheromone amount associated with a move
type rckt struct {
r, c, k int
t float64
}

func main() {
fmt.Println("Starting square: row", rStart, "column", cStart)
// initialize board
for r := 0; r < boardSize; r++ {
for c := 0; c < boardSize; c++ {
for k := 0; k < 8; k++ {
if _, _, ok := dest(r, c, k); ok {
tNet[(r*boardSize+c)*8+k] = 1e-6
}
}
}
}

// waitGroups for ant release clockwork
var start, reset sync.WaitGroup
tch := make(chan []rckt)

// create an ant for each square
for r := 0; r < boardSize; r++ {
for c := 0; c < boardSize; c++ {
go ant(r, c, &start, &reset, tch)
}
}

// accumulator for new pheromone amounts
tNew := make([]float64, nSquares*8)

// each iteration is a "cycle" as described in the paper
for {
// evaporate pheromones
for i := range tNet {
tNet[i] *= .75
}

reset.Add(nSquares) // number of ants to release
start.Done() // release them
reset.Wait() // wait for them to begin searching
start.Add(1) // reset start signal for next cycle

// gather tours from ants
for i := 0; i < nSquares; i++ {
tour := <-tch
// watch for a complete tour from the specified starting square
if len(tour) == completeTour &&
tour[0].r == rStart-1 && tour[0].c == cStart-1 {

// task output: move sequence in a grid.
seq := make([]int, nSquares)
for i, sq := range tour {
seq[sq.r*boardSize+sq.c] = i + 1
}
last := tour[len(tour)-1]
r, c, _ := dest(last.r, last.c, last.k)
seq[r*boardSize+c] = nSquares
fmt.Println("Move sequence:")
for r := 0; r < boardSize; r++ {
for c := 0; c < boardSize; c++ {
fmt.Printf(" %3d", seq[r*boardSize+c])
}
fmt.Println()
}
return // task only requires finding a single tour
}
// accumulate pheromone amounts from all ants
for _, move := range tour {
tNew[(move.r*boardSize+move.c)*8+move.k] += move.t
}
}

// update pheromone amounts on network, reset accumulator
for i, tn := range tNew {
tNet[i] += tn
tNew[i] = 0
}
}
}

type square struct {
r, c int
}

func ant(r, c int, start, reset *sync.WaitGroup, tourCh chan []rckt) {
rnd := rand.New(rand.NewSource(time.Now().UnixNano()))
tabu := make([]square, nSquares)
moves := make([]rckt, nSquares)
unexp := make([]rckt, 8)
tabu[0].r = r
tabu[0].c = c

for {
// cycle initialization
moves = moves[:0]
tabu = tabu[:1]
r := tabu[0].r
c := tabu[0].c

// wait for start signal
start.Wait()
reset.Done()

for {
// choose next move
unexp = unexp[:0]
var tSum float64
findU:
for k := 0; k < 8; k++ {
dr, dc, ok := dest(r, c, k)
if !ok {
continue
}
for _, t := range tabu {
if t.r == dr && t.c == dc {
continue findU
}
}
tk := tNet[(r*boardSize+c)*8+k]
tSum += tk
// note: dest r, c stored here
unexp = append(unexp, rckt{dr, dc, k, tk})
}
if len(unexp) == 0 {
break // no moves
}
rn := rnd.Float64() * tSum
var move rckt
for _, move = range unexp {
if rn <= move.t {
break
}
rn -= move.t
}

// move to new square
move.r, r = r, move.r
move.c, c = c, move.c
tabu = append(tabu, square{r, c})
moves = append(moves, move)
}

// compute pheromone amount to leave
for i := range moves {
moves[i].t = float64(len(moves)-i) / float64(completeTour-i)
}

tourCh <- moves
}
}

Output:

Starting square:  row 2 column 3
Move sequence:
64  33  36   3  54  49  38  51
35   4   1  30  37  52  55  48
32  63  34  53   2  47  50  39
5  18  31  46  29  20  13  56
62  27  44  19  14  11  40  21
17   6  15  28  45  22  57  12
26  61   8  43  24  59  10  41
7  16  25  60   9  42  23  58

{-# LANGUAGE TupleSections #-}

import Data.List (minimumBy, (\\), intercalate, sort)
import Data.Ord (comparing)
import Data.Char (ord, chr)
import Data.Bool (bool)

type Square = (Int, Int)

knightTour :: [Square] -> [Square]
knightTour moves
| null possibilities = reverse moves
| otherwise = knightTour \$ newSquare : moves
where
newSquare = minimumBy (comparing (length . findMoves)) possibilities
possibilities = findMoves \$ head moves
findMoves = (\\ moves) . knightOptions

knightOptions :: Square -> [Square]
knightOptions (x, y) =
knightMoves >>=
(\(i, j) ->
let a = x + i
b = y + j
in bool [] [(a, b)] (onBoard a && onBoard b))

knightMoves :: [(Int, Int)]
knightMoves =
let deltas = [id, negate] <*> [1, 2]
in deltas >>=
(\i -> deltas >>= (bool [] . return . (i, )) <*> ((abs i /=) . abs))

onBoard :: Int -> Bool
onBoard = (&&) . (0 <) <*> (9 >)

-- TEST ---------------------------------------------------
startPoint = "e5"

algebraic :: (Int, Int) -> String
algebraic (x, y) = [chr (x + 96), chr (y + 48)]

main :: IO ()
main =
printTour \$
algebraic <\$> knightTour [(\[x, y] -> (ord x - 96, ord y - 48)) startPoint]
where
printTour [] = return ()
printTour tour = do
putStrLn \$ intercalate " -> " \$ take 8 tour
printTour \$ drop 8 tour
Output:
e5 -> f7 -> h8 -> g6 -> h4 -> g2 -> e1 -> f3
g1 -> h3 -> g5 -> h7 -> f8 -> d7 -> b8 -> a6
b4 -> a2 -> c1 -> d3 -> b2 -> d1 -> f2 -> h1
g3 -> h5 -> g7 -> e8 -> f6 -> g8 -> h6 -> g4
h2 -> f1 -> e3 -> f5 -> e7 -> c8 -> a7 -> c6
d8 -> b7 -> a5 -> b3 -> a1 -> c2 -> d4 -> e2
f4 -> e6 -> c5 -> a4 -> b6 -> a8 -> c7 -> d5
c3 -> e4 -> d6 -> b5 -> a3 -> b1 -> d2 -> c4

## Icon and Unicon

This implements Warnsdorff's algorithm using unordered sets.

• The board must be square (it has only been tested on 8x8 and 7x7). Currently the maximum size board (due to square notation) is 26x26.
• Tie breaking is selectable with 3 variants supplied (first in list, random, and Roth's distance heuristic).
• A debug log can be generated showing the moves and choices considered for tie breaking.

The algorithm doesn't always generate a complete tour.

procedure main(A)
ShowTour(KnightsTour(Board(8)))
end

procedure KnightsTour(B,sq,tbrk,debug) #: Warnsdorff’s algorithm

/B := Board(8) # create 8x8 board if none given
/sq := ?B.files || ?B.ranks # random initial position (default)
sq2fr(sq,B) # validate initial sq
if type(tbrk) == "procedure" then
B.tiebreak := tbrk # override tie-breaker
if \debug then write("Debug log : move#, move : (accessibility) choices")

choices := [] # setup to track moves and choices
every (movesto := table())[k := key(B.movesto)] := copy(B.movesto[k])

B.tour := [] # new tour
repeat {
put(B.tour,sq) # record move

ac := 9 # accessibility counter > maximum
while get(choices) # empty choices for tiebreak
every delete(movesto[nextsq := !movesto[sq]],sq) do { # make sq unavailable
if ac >:= *movesto[nextsq] then # reset to lower accessibility count
while get(choices) # . re-empty choices
if ac = *movesto[nextsq] then
put(choices,nextsq) # keep least accessible sq and any ties
}

if \debug then { # move#, move, (accessibility), choices
writes(sprintf("%d. %s : (%d) ",*B.tour,sq,ac))
every writes(" ",!choices|"\n")
}
sq := B.tiebreak(choices,B) | break # choose next sq until out of choices
}
return B
end

procedure RandomTieBreaker(S,B) # random choice
return ?S
end

procedure FirstTieBreaker(S,B) # first one in the list
return !S
end

procedure RothTieBreaker(S,B) # furthest from the center
if *S = 0 then fail # must fail if []
every fr := sq2fr(s := !S,B) do {
d := sqrt(abs(fr[1]-1 - (B.N-1)*0.5)^2 + abs(fr[2]-1 - (B.N-1)*0.5)^2)
if (/md := d) | ( md >:= d) then msq := s # save sq
}
return msq
end

record board(N,ranks,files,movesto,tiebreak,tour) # structure for board

procedure Board(N) #: create board
N := *&lcase >=( 0 < integer(N)) | stop("N=",image(N)," is out of range.")
B := board(N,[],&lcase[1+:N],table(),RandomTieBreaker) # setup
every put(B.ranks,N to 1 by -1) # add rank #s
every sq := !B.files || !B.ranks do # for each sq add
every insert(B.movesto[sq] := set(), KnightMoves(sq,B)) # moves to next sq
return B
end

procedure sq2fr(sq,B) #: return numeric file & rank
f := find(sq[1],B.files) | runerr(205,sq)
r := integer(B.ranks[sq[2:0]]) | runerr(205,sq)
return [f,r]
end

procedure KnightMoves(sq,B) #: generate all Kn accessible moves from sq
fr := sq2fr(sq,B)
every ( i := -2|-1|1|2 ) & ( j := -2|-1|1|2 ) do
if (abs(i)~=abs(j)) & (0<(ri:=fr[2]+i)<=B.N) & (0<(fj:=fr[1]+j)<=B.N) then
suspend B.files[fj]||B.ranks[ri]
end

procedure ShowTour(B) #: show the tour
write("Board size = ",B.N)
write("Tour length = ",*B.tour)
write("Tie Breaker = ",image(B.tiebreak))

every !(squares := list(B.N)) := list(B.N,"-")
every fr := sq2fr(B.tour[m := 1 to *B.tour],B) do
squares[fr[2],fr[1]] := m

every (hdr1 := " ") ||:= right(!B.files,3)
every (hdr2 := " +") ||:= repl((1 to B.N,"-"),3) | "-+"

every write(hdr1|hdr2)
every r := 1 to B.N do {
writes(right(B.ranks[r],3)," |")
every writes(right(squares[r,f := 1 to B.N],3))
write(" |",right(B.ranks[r],3))
}
every write(hdr2|hdr1|&null)
end

The following can be used when debugging to validate the board structure and to image the available moves on the board.

procedure DumpBoard(B)  #: Dump Board internals
write("Board size=",B.N)
write("Available Moves at start of tour:", ImageMovesTo(B.movesto))
end

procedure ImageMovesTo(movesto) #: image of available moves
every put(K := [],key(movesto))
every (s := "\n") ||:= (k := !sort(K)) || " : " do
every s ||:= " " || (!sort(movesto[k])|"\n")
return s
end

Sample output:
Board size = 8
Tour length = 64
Tie Breaker = procedure RandomTieBreaker
a  b  c  d  e  f  g  h
+-------------------------+
8 | 53 10 29 26 55 12 31 16 |  8
7 | 28 25 54 11 30 15 48 13 |  7
6 |  9 52 27 62 47 56 17 32 |  6
5 | 24 61 38 51 36 45 14 49 |  5
4 | 39  8 63 46 57 50 33 18 |  4
3 | 64 23 60 37 42 35 44  3 |  3
2 |  7 40 21 58  5  2 19 34 |  2
1 | 22 59  6 41 20 43  4  1 |  1
+-------------------------+
a  b  c  d  e  f  g  h
Two 7x7 boards:
Board size = 7
Tour length = 33
Tie Breaker = procedure RandomTieBreaker
a  b  c  d  e  f  g
+----------------------+
7 | 33  4 15  - 29  6 17 |  7
6 | 14  - 30  5 16  - 28 |  6
5 |  3 32  -  -  - 18  7 |  5
4 |  - 13  - 31  - 27  - |  4
3 | 23  2  -  -  -  8 19 |  3
2 | 12  - 24 21 10  - 26 |  2
1 |  1 22 11  - 25 20  9 |  1
+----------------------+
a  b  c  d  e  f  g

Board size = 7
Tour length = 49
Tie Breaker = procedure RothTieBreaker
a  b  c  d  e  f  g
+----------------------+
7 | 35 14 21 46  7 12  9 |  7
6 | 20 49 34 13 10 23  6 |  6
5 | 15 36 45 22 47  8 11 |  5
4 | 42 19 48 33 40  5 24 |  4
3 | 37 16 41 44 27 32 29 |  3
2 | 18 43  2 39 30 25  4 |  2
1 |  1 38 17 26  3 28 31 |  1
+----------------------+
a  b  c  d  e  f  g

## J

Solution:
The Knight's tour essay on the Jwiki shows a couple of solutions including one using Warnsdorffs algorithm.

NB. knight moves for each square of a (y,y) board
t=. (>,{;~i.y) +"1/ _2]\2 1 2 _1 1 2 1 _2 _1 2 _1 _2 _2 1 _2 _1
(*./"1 t e. i.y) <@#"1 y#.t
)

M=. >kmoves y
p=. k=. 0
b=. 1 \$~ *:y
for. i.<:*:y do.
b=. 0 k}b
p=. p,k=. ((i.<./) +/"1 b{~j{M){j=. ({&b # ]) k{M
end.
assert. ~:p
(,~y)\$/:p
)

Example Use:

ktourw 8    NB. solution for an 8 x 8 board
0 25 14 23 28 49 12 31
15 22 27 50 13 30 63 48
26 1 24 29 62 59 32 11
21 16 51 58 43 56 47 60
2 41 20 55 52 61 10 33
17 38 53 42 57 44 7 46
40 3 36 19 54 5 34 9
37 18 39 4 35 8 45 6

9!:37]0 64 4 4 NB. truncate lines longer than 64 characters and only show first and last four lines

ktourw 202 NB. 202x202 board -- this implementation failed for 200 and 201
0 401 414 405 398 403 424 417 396 419 43...
413 406 399 402 425 416 397 420 439 430 39...
400 1 426 415 404 423 448 429 418 437 4075...
409 412 407 446 449 428 421 440 40739 40716 43...
...
550 99 560 569 9992 779 786 773 10002 9989 78...
555 558 553 778 563 570 775 780 785 772 1000...
100 551 556 561 102 777 572 771 104 781 57...
557 554 101 552 571 562 103 776 573 770 10...

## Java

Works with: Java version 7
import java.util.*;

public class KnightsTour {
private final static int base = 12;
private final static int[][] moves = {{1,-2},{2,-1},{2,1},{1,2},{-1,2},
{-2,1},{-2,-1},{-1,-2}};
private static int[][] grid;
private static int total;

public static void main(String[] args) {
grid = new int[base][base];
total = (base - 4) * (base - 4);

for (int r = 0; r < base; r++)
for (int c = 0; c < base; c++)
if (r < 2 || r > base - 3 || c < 2 || c > base - 3)
grid[r][c] = -1;

int row = 2 + (int) (Math.random() * (base - 4));
int col = 2 + (int) (Math.random() * (base - 4));

grid[row][col] = 1;

if (solve(row, col, 2))
printResult();
else System.out.println("no result");

}

private static boolean solve(int r, int c, int count) {
if (count > total)
return true;

List<int[]> nbrs = neighbors(r, c);

if (nbrs.isEmpty() && count != total)
return false;

Collections.sort(nbrs, new Comparator<int[]>() {
public int compare(int[] a, int[] b) {
return a[2] - b[2];
}
});

for (int[] nb : nbrs) {
r = nb[0];
c = nb[1];
grid[r][c] = count;
if (!orphanDetected(count, r, c) && solve(r, c, count + 1))
return true;
grid[r][c] = 0;
}

return false;
}

private static List<int[]> neighbors(int r, int c) {
List<int[]> nbrs = new ArrayList<>();

for (int[] m : moves) {
int x = m[0];
int y = m[1];
if (grid[r + y][c + x] == 0) {
int num = countNeighbors(r + y, c + x);
nbrs.add(new int[]{r + y, c + x, num});
}
}
return nbrs;
}

private static int countNeighbors(int r, int c) {
int num = 0;
for (int[] m : moves)
if (grid[r + m[1]][c + m[0]] == 0)
num++;
return num;
}

private static boolean orphanDetected(int cnt, int r, int c) {
if (cnt < total - 1) {
List<int[]> nbrs = neighbors(r, c);
for (int[] nb : nbrs)
if (countNeighbors(nb[0], nb[1]) == 0)
return true;
}
return false;
}

private static void printResult() {
for (int[] row : grid) {
for (int i : row) {
if (i == -1) continue;
System.out.printf("%2d ", i);
}
System.out.println();
}
}
}
34 17 20  3 36  7 22  5
19  2 35 40 21  4 37  8
16 33 18 51 44 39  6 23
1 50 43 46 41 56  9 38
32 15 54 61 52 45 24 57
49 62 47 42 55 60 27 10
14 31 64 53 12 29 58 25
63 48 13 30 59 26 11 28

Works with: Java version 8

package com.knight.tour;
import java.util.ArrayList;
import java.util.Collections;
import java.util.Comparator;
import java.util.List;

public class KT {

private int baseSize = 12; // virtual board size including unreachable out-of-board nodes. i.e. base 12 = 8X8 board
int actualBoardSize = baseSize - 4;
private static final int[][] moves = { { 1, -2 }, { 2, -1 }, { 2, 1 }, { 1, 2 }, { -1, 2 }, { -2, 1 }, { -2, -1 },
{ -1, -2 } };
private static int[][] grid;
private static int totalNodes;
private ArrayList<int[]> travelledNodes = new ArrayList<>();
public KT(int baseNumber) {
this.baseSize = baseNumber;
this.actualBoardSize = baseSize - 4;
}

public static void main(String[] args) {
new KT(12).tour(); // find a solution for 8X8 board
// new KT(24).tour(); // then for 20X20 board
// new KT(104).tour(); // then for 100X100 board
}

private void tour() {
totalNodes = actualBoardSize * actualBoardSize;
travelledNodes.clear();
grid = new int[baseSize][baseSize];
for (int r = 0; r < baseSize; r++)
for (int c = 0; c < baseSize; c++) {
if (r < 2 || r > baseSize - 3 || c < 2 || c > baseSize - 3) {
grid[r][c] = -1; // mark as out-of-board nodes
} else {
grid[r][c] = 0; // nodes within chess board.
}
}
// start from a random node
int startRow = 2 + (int) (Math.random() * actualBoardSize);
int startCol = 2 + (int) (Math.random() * actualBoardSize);
int[] start = { startRow, startCol, 0, 1 };
grid[startRow][startCol] = 1; // mark the first traveled node
travelledNodes.add(start); // add to partial solution chain, which will only have one node.

// Start traveling forward
autoKnightTour(start, 2);
}

// non-backtracking touring methods. Re-chain the partial solution when all neighbors are traveled to avoid back-tracking.
private void autoKnightTour(int[] start, int nextCount) {
List<int[]> nbrs = neighbors(start[0], start[1]);
if (nbrs.size() > 0) {
Collections.sort(nbrs, new Comparator<int[]>() {
public int compare(int[] a, int[] b) {
return a[2] - b[2];
}
}); // sort the list
int[] next = nbrs.get(0); // the one with the less available neighbors - Warnsdorff's algorithm
next[3] = nextCount;
grid[next[0]][next[1]] = nextCount;
if (travelledNodes.size() == totalNodes) {
System.out.println("Found a path for " + actualBoardSize + " X " + actualBoardSize + " chess board.");
StringBuilder sb = new StringBuilder();
sb.append(System.lineSeparator());
for (int idx = 0; idx < travelledNodes.size(); idx++) {
int[] item = travelledNodes.get(idx);
sb.append("->(" + (item[0] - 2) + "," + (item[1] - 2) + ")");
if ((idx + 1) % 15 == 0) {
sb.append(System.lineSeparator());
}
}
System.out.println(sb.toString() + "\n");
} else { // continuing the travel
autoKnightTour(next, ++nextCount);
}
} else { // no travelable neighbors next - need to rechain the partial chain
int[] last = travelledNodes.get(travelledNodes.size() - 1);
travelledNodes = reChain(travelledNodes);
if (travelledNodes.get(travelledNodes.size() - 1).equals(last)) {
travelledNodes = reChain(travelledNodes);
if (travelledNodes.get(travelledNodes.size() - 1).equals(last)) {
System.out.println("Re-chained twice but no travllable node found. Quiting...");
} else {
int[] end = travelledNodes.get(travelledNodes.size() - 1);
autoKnightTour(end, nextCount);
}
} else {
int[] end = travelledNodes.get(travelledNodes.size() - 1);
autoKnightTour(end, nextCount);
}
}
}

List<int[]> candidates = neighborsInChain(last[0], last[1]);
int cutIndex;
int[] randomPicked = candidates.get((int) Math.random() * candidates.size());
cutIndex = grid[randomPicked[0]][randomPicked[1]] - 1;
ArrayList<int[]> result = new ArrayList<int[]>(); //create empty list to copy already traveled nodes to
for (int k = 0; k <= cutIndex; k++) {
}
for (int j = alreadyTraveled.size() - 1; j > cutIndex; j--) {
}
return result; // re-chained partial solution with different end node
}

private List<int[]> neighborsInChain(int r, int c) {
List<int[]> nbrs = new ArrayList<>();
for (int[] m : moves) {
int x = m[0];
int y = m[1];
if (grid[r + y][c + x] > 0 && grid[r + y][c + x] != grid[r][c] - 1) {
int num = countNeighbors(r + y, c + x);
nbrs.add(new int[] { r + y, c + x, num, 0 });
}
}
return nbrs;
}

private static List<int[]> neighbors(int r, int c) {
List<int[]> nbrs = new ArrayList<>();
for (int[] m : moves) {
int x = m[0];
int y = m[1];
if (grid[r + y][c + x] == 0) {
int num = countNeighbors(r + y, c + x);
nbrs.add(new int[] { r + y, c + x, num, 0 }); // not-traveled neighbors and number of their neighbors
}
}
return nbrs;

}

private List<int[]> extendableNeighbors(List<int[]> neighbors) {
List<int[]> nbrs = new ArrayList<>();
for (int[] node : neighbors) {
if (node[2] > 0)
}
return nbrs;
}

private static int countNeighbors(int r, int c) {
int num = 0;
for (int[] m : moves) {
if (grid[r + m[1]][c + m[0]] == 0) {
num++;
}
}
return num;
}
}

Found a path for 8 X 8 chess board.

->(2,1)->(0,0)->(1,2)->(0,4)->(1,6)->(3,7)->(5,6)->(7,7)->(6,5)->(5,7)->(7,6)->(6,4)->(7,2)->(6,0)->(4,1)
->(2,0)->(0,1)->(1,3)->(0,5)->(1,7)->(3,6)->(2,4)->(0,3)->(1,1)->(3,0)->(2,2)->(1,0)->(0,2)->(1,4)->(0,6)
->(2,7)->(1,5)->(0,7)->(2,6)->(4,7)->(6,6)->(4,5)->(3,3)->(2,5)->(4,6)->(6,7)->(7,5)->(5,4)->(3,5)->(2,3)
->(4,4)->(3,2)->(4,0)->(5,2)->(7,3)->(6,1)->(5,3)->(3,4)->(4,2)->(6,3)->(7,1)->(5,0)->(3,1)->(4,3)->(5,5)
->(7,4)->(6,2)->(7,0)->(5,1)

## JavaScript

### Procedural

Using Warnsdorff rule and Backtracking.

You can test it here.

class KnightTour {
constructor() {
this.width = 856;
this.height = 856;
this.cellCount = 8;
this.size = 0;
this.knightPiece = "\u2658";
this.knightPos = {
x: 0,
y: 0
};
this.ctx = null;
this.step = this.width / this.cellCount;
this.lastTime = 0;
this.wait;
this.delay;
this.success;
this.jumps;
this.directions = [];
this.visited = [];
this.path = [];
this.startHtml();
});
this.init();
this.drawBoard();
}

drawBoard() {
let a = false, xx, yy;
for (let y = 0; y < this.cellCount; y++) {
for (let x = 0; x < this.cellCount; x++) {
if (a) {
this.ctx.fillStyle = "#607db8";
} else {
this.ctx.fillStyle = "#aecaf0";
}
a = !a;
xx = x * this.step;
yy = y * this.step;
this.ctx.fillRect(xx, yy, xx + this.step, yy + this.step);
}
if (!(this.cellCount & 1)) a = !a;
}
if (this.path.length) {
const s = this.step >> 1;
this.ctx.lineWidth = 3;
this.ctx.fillStyle = "black";
this.ctx.beginPath();
this.ctx.moveTo(this.step * this.knightPos.x + s, this.step * this.knightPos.y + s);
let a, b, v = this.path.length - 1;
for (; v > -1; v--) {
a = this.path[v].pos.x * this.step + s;
b = this.path[v].pos.y * this.step + s;
this.ctx.lineTo(a, b);
this.ctx.fillRect(a - 5, b - 5, 10, 10);
}
this.ctx.stroke();
}
}

createMoves(pos) {
const possibles = [];
let x = 0,
y = 0,
m = 0,
l = this.directions.length;
for (; m < l; m++) {
x = pos.x + this.directions[m].x;
y = pos.y + this.directions[m].y;
if (x > -1 && x < this.cellCount && y > -1 && y < this.cellCount && !this.visited[x + y * this.cellCount]) {
possibles.push({
x,
y
})
}
}
return possibles;
}

warnsdorff(pos) {
const possibles = this.createMoves(pos);
if (possibles.length < 1) return [];
const moves = [];
for (let p = 0, l = possibles.length; p < l; p++) {
let ps = this.createMoves(possibles[p]);
moves.push({
len: ps.length,
pos: possibles[p]
});
}
moves.sort((a, b) => {
return b.len - a.len;
});
return moves;
}

startHtml() {
this.cellCount = parseInt(document.getElementById("cellCount").value);
this.size = Math.floor(this.width / this.cellCount)
this.wait = this.delay = parseInt(document.getElementById("delay").value);
this.step = this.width / this.cellCount;
this.ctx.font = this.size + "px Arial";
document.getElementById("log").innerText = "";
document.getElementById("path").innerText = "";
this.path = [];
this.jumps = 1;
this.success = true;
this.visited = [];
const cnt = this.cellCount * this.cellCount;
for (let a = 0; a < cnt; a++) {
this.visited.push(false);
}
const kx = parseInt(document.getElementById("knightx").value),
ky = parseInt(document.getElementById("knighty").value);
this.knightPos = {
x: (kx > this.cellCount || kx < 0) ? Math.floor(Math.random() * this.cellCount) : kx,
y: (ky > this.cellCount || ky < 0) ? Math.floor(Math.random() * this.cellCount) : ky
};
this.mainLoop = (time = 0) => {
const dif = time - this.lastTime;
this.lastTime = time;
this.wait -= dif;
if (this.wait > 0) {
requestAnimationFrame(this.mainLoop);
return;
}
this.wait = this.delay;
let moves;
if (this.success) {
moves = this.warnsdorff(this.knightPos);
} else {
if (this.path.length > 0) {
const path = this.path[this.path.length - 1];
moves = path.m;
if (moves.length < 1) this.path.pop();
this.knightPos = path.pos
this.visited[this.knightPos.x + this.knightPos.y * this.cellCount] = false;
this.jumps--;
this.wait = this.delay;
} else {
document.getElementById("log").innerText = "Can't find a solution!";
return;
}
}
this.drawBoard();
const ft = this.step - (this.step >> 3);
this.ctx.fillStyle = "#000";
this.ctx.fillText(this.knightPiece, this.knightPos.x * this.step, this.knightPos.y * this.step + ft);
if (moves.length < 1) {
if (this.jumps === this.cellCount * this.cellCount) {
document.getElementById("log").innerText = "Tour finished!";
let str = "";
for (let z of this.path) {
str += `\${1 + z.pos.x + z.pos.y * this.cellCount}, `;
}
str += `\${1 + this.knightPos.x + this.knightPos.y * this.cellCount}`;
document.getElementById("path").innerText = str;
return;
} else {
this.success = false;
}
} else {
this.visited[this.knightPos.x + this.knightPos.y * this.cellCount] = true;
const move = moves.pop();
this.path.push({
pos: this.knightPos,
m: moves
});
this.knightPos = move.pos
this.success = true;
this.jumps++;
}
requestAnimationFrame(this.mainLoop);
};
this.mainLoop();
}

init() {
const canvas = document.createElement("canvas");
canvas.id = "cv";
canvas.width = this.width;
canvas.height = this.height;
this.ctx = canvas.getContext("2d");
document.getElementById("out").appendChild(canvas);
this.directions = [{
x: -1,
y: -2
}, {
x: -2,
y: -1
}, {
x: 1,
y: -2
}, {
x: 2,
y: -1
},
{
x: -1,
y: 2
}, {
x: -2,
y: 1
}, {
x: 1,
y: 2
}, {
x: 2,
y: 1
}
];
}
}
new KnightTour();

To test it, you'll need an index.html

<!DOCTYPE html>
<html>
<meta charset="UTF-8">
<title>Knight's Tour</title>
<link rel="stylesheet" type="text/css" media="screen" href="style.css" />
<body>
<div id='out'></div>
<div id='ctrls'>
<span>Cells: </span><input id="cellCount" value="8" type="number" max="250" min="5"><br />
<span>Delay: </span><input id="delay" value="500" type="number" max="2000" min="0"><br />
<span>Knight X: </span><input id="knightx" value="-1" type="number" max="250" min="-1"><br />
<span>Knight Y: </span><input id="knighty" value="-1" type="number" max="250" min="-1"><br />
<button id="start">Start</button>
<div id='log'></div>
<div id="path"></div>
</div>
<script src="tour_bt.js" type="module"></script>
</body>
</html>

And a style.css

body {
font-family: verdana;
color: white;
font-size: 36px;
background-color: #001f33
}
button {
width: 100%;
height: 40px;
margin: 20px 0px 20px 0px;
font-size: 28px
}
canvas {
border: 4px solid #000;
margin: 40px;
}
#out {
float: left;
}
#ctrls {
margin-top: 40px;
text-align: left;
width: 280px;
line-height: 40px;
float: left;
}
#ctrls input {
float: right;
width: 80px;
height: 24px;
margin-top: 6px;
font-size: 22px;
}
#path {
margin-top: 10px;
font-size: 12px;
line-height: 16px;
}

### Functional

A composition of values, drawing on generic abstractions:

(() => {
'use strict';

// knightsTour :: Int -> [(Int, Int)] -> [(Int, Int)]
const knightsTour = rowLength => moves => {
const go = path => {
const
findMoves = xy => difference(knightMoves(xy), path),
warnsdorff = minimumBy(
comparing(compose(length, findMoves))
),
options = findMoves(path[0]);
return 0 < options.length ? (
go([warnsdorff(options)].concat(path))
) : reverse(path);
};

// board :: [[(Int, Int)]]
const board = concatMap(
col => concatMap(
row => [
[col, row]
],
enumFromTo(1, rowLength)),
enumFromTo(1, rowLength)
);

// knightMoves :: (Int, Int) -> [(Int, Int)]
const knightMoves = ([x, y]) =>
concatMap(
([dx, dy]) => {
const ab = [x + dx, y + dy];
return elem(ab, board) ? (
[ab]
) : [];
}, [
[-2, -1],
[-2, 1],
[-1, -2],
[-1, 2],
[1, -2],
[1, 2],
[2, -1],
[2, 1]
]
);
return go(moves);
};

// TEST -----------------------------------------------
// main :: IO()
const main = () => {

// boardSize :: Int
const boardSize = 8;

// tour :: [(Int, Int)]
const tour = knightsTour(boardSize)(
[fromAlgebraic('e5')]
);

// report :: String
const report = '(Board size ' +
boardSize + '*' + boardSize + ')\n\n' +
'Route: \n\n' +
showRoute(boardSize)(tour) + '\n\n' +
'Coverage and order: \n\n' +
showCoverage(boardSize)(tour) + '\n\n';
return (
console.log(report),
report
);
}

// DISPLAY --------------------------------------------

// algebraic :: (Int, Int) -> String
const algebraic = ([x, y]) =>
chr(x + 96) + y.toString();

// fromAlgebraic :: String -> (Int, Int)
const fromAlgebraic = s =>
2 <= s.length ? (
[ord(s[0]) - 96, parseInt(s.slice(1))]
) : undefined;

// showCoverage :: Int -> [(Int, Int)] -> String
const showCoverage = rowLength => xys => {
const
intMax = xys.length,
w = 1 + intMax.toString().length
return unlines(map(concat,
chunksOf(
rowLength,
map(composeList([justifyRight(w, ' '), str, fst]),
sortBy(
mappendComparing([
compose(fst, snd),
compose(snd, snd)
]),
zip(enumFromTo(1, intMax), xys)
)
)
)
));
};

// showRoute :: Int -> [(Int, Int)] -> String
const showRoute = rowLength => xys => {
const w = 1 + rowLength.toString().length;
return unlines(map(
xs => xs.join(' -> '),
chunksOf(
rowLength,
map(compose(justifyRight(w, ' '), algebraic), xys)
)
));
};

// GENERIC FUNCTIONS ----------------------------------

// Tuple (,) :: a -> b -> (a, b)
const Tuple = (a, b) => ({
type: 'Tuple',
'0': a,
'1': b,
length: 2
});

// chr :: Int -> Char
const chr = x => String.fromCodePoint(x);

// chunksOf :: Int -> [a] -> [[a]]
const chunksOf = (n, xs) =>
enumFromThenTo(0, n, xs.length - 1)
.reduce(
(a, i) => a.concat([xs.slice(i, (n + i))]),
[]
);

// compare :: a -> a -> Ordering
const compare = (a, b) =>
a < b ? -1 : (a > b ? 1 : 0);

// comparing :: (a -> b) -> (a -> a -> Ordering)
const comparing = f =>
(x, y) => {
const
a = f(x),
b = f(y);
return a < b ? -1 : (a > b ? 1 : 0);
};

// compose (<<<) :: (b -> c) -> (a -> b) -> a -> c
const compose = (f, g) => x => f(g(x));

// composeList :: [(a -> a)] -> (a -> a)
const composeList = fs =>
x => fs.reduceRight((a, f) => f(a), x, fs);

// concat :: [[a]] -> [a]
// concat :: [String] -> String
const concat = xs =>
0 < xs.length ? (() => {
const unit = 'string' !== typeof xs[0] ? (
[]
) : '';
return unit.concat.apply(unit, xs);
})() : [];

// concatMap :: (a -> [b]) -> [a] -> [b]
const concatMap = (f, xs) =>
xs.reduce((a, x) => a.concat(f(x)), []);

// difference :: Eq a => [a] -> [a] -> [a]
const difference = (xs, ys) => {
const s = new Set(ys.map(str));
return xs.filter(x => !s.has(str(x)));
};

// elem :: Eq a => a -> [a] -> Bool
const elem = (x, xs) => xs.some(eq(x))

// enumFromThenTo :: Int -> Int -> Int -> [Int]
const enumFromThenTo = (x1, x2, y) => {
const d = x2 - x1;
return Array.from({
length: Math.floor(y - x2) / d + 2
}, (_, i) => x1 + (d * i));
};

// enumFromTo :: Int -> Int -> [Int]
const enumFromTo = (m, n) =>
Array.from({
length: 1 + n - m
}, (_, i) => m + i);

// eq (==) :: Eq a => a -> a -> Bool
const eq = a => b => {
const t = typeof a;
return t !== typeof b ? (
false
) : 'object' !== t ? (
'function' !== t ? (
a === b
) : a.toString() === b.toString()
) : (() => {
const kvs = Object.entries(a);
return kvs.length !== Object.keys(b).length ? (
false
) : kvs.every(([k, v]) => eq(v)(b[k]));
})();
};

// fst :: (a, b) -> a
const fst = tpl => tpl[0];

// justifyRight :: Int -> Char -> String -> String
const justifyRight = (n, cFiller) => s =>
n > s.length ? (
) : s;

// length :: [a] -> Int
const length = xs =>
(Array.isArray(xs) || 'string' === typeof xs) ? (
xs.length
) : Infinity;

// map :: (a -> b) -> [a] -> [b]
const map = (f, xs) =>
(Array.isArray(xs) ? (
xs
) : xs.split('')).map(f);

// mappendComparing :: [(a -> b)] -> (a -> a -> Ordering)
const mappendComparing = fs =>
(x, y) => fs.reduce(
(ordr, f) => (ordr || compare(f(x), f(y))),
0
);

// minimumBy :: (a -> a -> Ordering) -> [a] -> a
const minimumBy = f => xs =>
xs.reduce((a, x) => undefined === a ? x : (
0 > f(x, a) ? x : a
), undefined);

// ord :: Char -> Int
const ord = c => c.codePointAt(0);

// reverse :: [a] -> [a]
const reverse = xs =>
'string' !== typeof xs ? (
xs.slice(0).reverse()
) : xs.split('').reverse().join('');

// snd :: (a, b) -> b
const snd = tpl => tpl[1];

// sortBy :: (a -> a -> Ordering) -> [a] -> [a]
const sortBy = (f, xs) =>
xs.slice()
.sort(f);

// str :: a -> String
const str = x => x.toString();

// take :: Int -> [a] -> [a]
// take :: Int -> String -> String
const take = (n, xs) =>
xs.slice(0, n);

// unlines :: [String] -> String
const unlines = xs => xs.join('\n');

// Use of `take` and `length` here allows for zipping with non-finite
// lists - i.e. generators like cycle, repeat, iterate.

// zip :: [a] -> [b] -> [(a, b)]
const zip = (xs, ys) => {
const lng = Math.min(length(xs), length(ys));
const bs = take(lng, ys);
return take(lng, xs).map((x, i) => Tuple(x, bs[i]));
};

// MAIN ---
return main();
})();
Output:
(Board size 8*8)

Route:

e5 -> d7 -> b8 -> a6 -> b4 -> a2 -> c1 -> b3
a1 -> c2 -> a3 -> b1 -> d2 -> f1 -> h2 -> g4
h6 -> g8 -> e7 -> c8 -> a7 -> c6 -> a5 -> b7
d8 -> f7 -> h8 -> g6 -> f8 -> h7 -> f6 -> e8
g7 -> h5 -> g3 -> h1 -> f2 -> d1 -> b2 -> a4
b6 -> a8 -> c7 -> b5 -> c3 -> d5 -> e3 -> c4
d6 -> e4 -> c5 -> d3 -> e1 -> g2 -> h4 -> f5
d4 -> e2 -> f4 -> e6 -> g5 -> f3 -> g1 -> h3

Coverage and order:

9  6 11 40 23  4 21 42
12 39  8  5 44 41 24  3
7 10 45 48 51 22 43 20
38 13 52 57 46 49  2 25
53 58 47 50  1 60 19 32
14 37 62 59 56 31 26 29
63 54 35 16 61 28 33 18
36 15 64 55 34 17 30 27

## Julia

Uses the Hidato puzzle solver module, which has its source code listed here in the Hadato task.

using .Hidato       # Note that the . here means to look locally for the module rather than in the libraries

const chessboard = """
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 """

const knightmoves = [[-2, -1], [-2, 1], [-1, -2], [-1, 2], [1, -2], [1, 2], [2, -1], [2, 1]]

board, maxmoves, fixed, starts = hidatoconfigure(chessboard)
printboard(board, " 0", " ")
hidatosolve(board, maxmoves, knightmoves, fixed, starts[1][1], starts[1][2], 1)
printboard(board)

Output:

0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0

1 12  9  6  3 14 17 20
10  7  2 13 18 21  4 15
31 28 11  8  5 16 19 22
64 25 32 29 36 23 48 45
33 30 27 24 49 46 37 58
26 63 52 35 40 57 44 47
53 34 61 50 55 42 59 38
62 51 54 41 60 39 56 43

## Kotlin

data class Square(val x : Int, val y : Int)

val board = Array(8 * 8, { Square(it / 8 + 1, it % 8 + 1) })
val axisMoves = arrayOf(1, 2, -1, -2)

fun <T> allPairs(a: Array<T>) = a.flatMap { i -> a.map { j -> Pair(i, j) } }

fun knightMoves(s : Square) : List<Square> {
val moves = allPairs(axisMoves).filter{ Math.abs(it.first) != Math.abs(it.second) }
fun onBoard(s : Square) = board.any {it == s}
return moves.map { Square(s.x + it.first, s.y + it.second) }.filter(::onBoard)
}

fun knightTour(moves : List<Square>) : List<Square> {
fun findMoves(s: Square) = knightMoves(s).filterNot { m -> moves.any { it == m } }
val newSquare = findMoves(moves.last()).minBy { findMoves(it).size }
return if (newSquare == null) moves else knightTour(moves + newSquare)
}

fun knightTourFrom(start : Square) = knightTour(listOf(start))

fun main(args : Array<String>) {
var col = 0
for ((x, y) in knightTourFrom(Square(1, 1))) {
System.out.print("\$x,\$y")
System.out.print(if (col == 7) "\n" else " ")
col = (col + 1) % 8
}
}
Output:
1,1 2,3 3,1 1,2 2,4 1,6 2,8 4,7
6,8 8,7 7,5 8,3 7,1 5,2 7,3 8,1
6,2 4,1 2,2 1,4 2,6 1,8 3,7 5,8
7,7 8,5 6,6 7,8 8,6 7,4 8,2 6,1
4,2 2,1 3,3 5,4 3,5 4,3 5,1 6,3
8,4 7,2 6,4 5,6 4,8 2,7 1,5 3,6
1,7 3,8 5,7 4,5 5,3 6,5 4,4 3,2
1,3 2,5 4,6 3,4 5,5 6,7 8,8 7,6

## Locomotive Basic

Influenced by the Python version, although computed tours are different.

10 mode 1:defint a-z
20 input "Board size: ",size
30 input "Start position: ",a\$
40 x=asc(mid\$(a\$,1,1))-96
50 y=val(mid\$(a\$,2,1))
60 dim play(size,size)
70 for q=1 to 8
90 next
100 data 2,1,1,2,-1,2,-2,1,-2,-1,-1,-2,1,-2,2,-1
110 pen 0:paper 1
120 for q=1 to size
130 locate 3*q+1,24-size
140 print chr\$(96+q);
150 locate 3*(size+1)+1,26-q
160 print using "#"; q;
170 next
180 pen 1:paper 0
190 ' main loop
200 n=n+1
210 play(x,y)=n
220 locate 3*x,26-y
230 print using "##"; n;
240 if n=size*size then call &bb06:end
250 nmov=100
260 for q=1 to 8
270 xc=x+dx(q)
280 yc=y+dy(q)
290 gosub 360
300 if nm<nmov then nmov=nm:qm=q
310 next
320 x=x+dx(qm)
330 y=y+dy(qm)
340 goto 200
350 ' find moves
360 if xc<1 or yc<1 or xc>size or yc>size then nm=1000:return
370 if play(xc,yc) then nm=2000:return
380 nm=0
390 for q2=1 to 8
400 xt=xc+dx(q2)
410 yt=yc+dy(q2)
420 if xt<1 or yt<1 or xt>size or yt>size then 460
430 if play(xt,yt) then 460
440 nm=nm+1
450 ' skip this move
460 next
470 return

## Lua

N = 8

moves = { {1,-2},{2,-1},{2,1},{1,2},{-1,2},{-2,1},{-2,-1},{-1,-2} }

function Move_Allowed( board, x, y )
if board[x][y] >= 8 then return false end

local new_x, new_y = x + moves[board[x][y]+1][1], y + moves[board[x][y]+1][2]
if new_x >= 1 and new_x <= N and new_y >= 1 and new_y <= N and board[new_x][new_y] == 0 then return true end

return false
end

board = {}
for i = 1, N do
board[i] = {}
for j = 1, N do
board[i][j] = 0
end
end

x, y = 1, 1

lst = {}
lst[1] = { x, y }

repeat
if Move_Allowed( board, x, y ) then
board[x][y] = board[x][y] + 1
x, y = x+moves[board[x][y]][1], y+moves[board[x][y]][2]
lst[#lst+1] = { x, y }
else
if board[x][y] >= 8 then
board[x][y] = 0
lst[#lst] = nil
if #lst == 0 then
print "No solution found."
os.exit(1)
end
x, y = lst[#lst][1], lst[#lst][2]
end
board[x][y] = board[x][y] + 1
end
until #lst == N^2

last = lst[1]
for i = 2, #lst do
print( string.format( "%s%d - %s%d", string.sub("ABCDEFGH",last[1],last[1]), last[2], string.sub("ABCDEFGH",lst[i][1],lst[i][1]), lst[i][2] ) )
last = lst[i]
end

## M2000 Interpreter

Function KnightTour\$(StartW=1, StartH=1){
def boolean swapH, swapV=True
if startW<=4 then swapH=true: StartW=8+1-StartW
if startH>4 then swapV=False: StartH=8+1-StartH
Let final=8*8, last=final-1, HighValue=final+1
Dim Board(1 to 8, 1 to 8), Moves(1 to 8, 1 to 8)=HighValue
f=stack:=1,2,3,4,5,6,7,8
if 8-StartW=2 and StartH=2 then stack f {shift 1,-8}
Function KnightMove(x,w,h) {
a=2:b=1:z=1:p=1
if x mod 2=1 then swap a,b
if x>2 then p-! : if x>4 then swap z, p : if x>6 then p-!
w+=z*a
h+=p*b
if w>=1 and w<=8 and h>=1 and h<=8 then =(w, h) else =(,)
}
For j=1 to 8 :For i=1 to 8
s=stack
For k=1 to 8
m=KnightMove(stackitem(f, k),i, j)
if len(m)>1 then Stack s {data m}
Next : Board(i,j)=s : Next
stack f {shift 1,-8}
Next
For i=1 to 8 :For j=1 to 8
s=Board(i, j)
if len(s)>2 then
so=queue
For k=1 to len(s)
m=stackitem(s, k)
Append so, Len(Board(m#val(0), m#val(1))) :=m
Next
sort ascending so as number
s=stack
stack s {for k=0 to len(so)-1:data so(k!):next}
Board(i,j)=s
end if
Next : Next
s= Board(StartW, StartH)
n=0
BackTrack=Stack
Moves=1
Moves(StartW, StartH)=1
Repeat
n++
While n>len(s) {
if Len(BackTrack)=0 then Print "Break", moves : Break
Moves--
m=stackitem(s, n)
Moves(m#val(0), m#val(1))=HighValue
n++
}
m=stackitem(s, n)
w=m#val(0)
h=m#val(1)
if Moves(w, h)>=Moves then
if Moves<last then
s1=Board(w, h) :ii=-1
for i=1 to len(s1){m1=stackitem(s1, i) :if Moves(m1#val(0),m1#val(1))>moves then ii=i-1 : exit
}
if ii>=0 then
Moves++
Moves(w,h)=Moves
Stack BackTrack {Push n, s}
s=s1: n=ii
end if
else
Moves++
Moves(w,h)=Moves
end if
end if
until Moves=final
Document export\$
Inventory Tour
letters=stack:="a","b","c","d","e","f","g","h"
f=stack:=1,2,3,4,5,6,7,8
if swapV Else stack f {Shift 1,-8}
if swapH then stack letters {Shift 1,-8}
For j=1 to 8:For i=1 to 8
Append Tour, Moves(i,j) :=stackitem\$(letters, i)+str\$(stackitem(f, j),"")
Next : Next
Sort ascending Tour as number
one=each(Tour)
While one {
export\$=Eval\$(one)
if not one^=last then export\$="->"
If (one^+1) mod 8=0 then
export\$={
}
End if
}
=export\$
}
Document ex\$
ex\$= {Knight's Tour from a1
}+KnightTour\$()+{Knight's Tour from h1
}+KnightTour\$(8,1)+{Knight's Tour from a8
}+KnightTour\$(1, 8)+{Knight's Tour from h8
}+KnightTour\$(8, 8)
Clipboard ex\$
Report ex\$

Output:
Knight's Tour from a1
a1->b3->a5->b7->d8->f7->h8->g6->
h4->g2->e1->c2->a3->b1->d2->f1->
h2->g4->h6->g8->e7->c8->a7->b5->
c7->a8->b6->a4->b2->d1->f2->h1->
g3->h5->g7->e8->f6->h7->f8->d7->
b8->a6->b4->a2->c1->e2->g1->h3->
g5->e6->f4->d3->c5->e4->c3->d5->
e3->c4->d6->f5->d4->f3->e5->c6
Knight's Tour from h1
h1->g3->h5->g7->e8->c7->a8->b6->
a4->b2->d1->f2->h3->g1->e2->c1->
a2->b4->a6->b8->d7->f8->h7->g5->
f7->h8->g6->h4->g2->e1->c2->a1->
b3->a5->b7->d8->c6->a7->c8->e7->
g8->h6->g4->h2->f1->d2->b1->a3->
b5->d6->c4->e3->f5->d4->f3->e5->
d3->f4->e6->c5->e4->c3->d5->f6
Knight's Tour from a8
a8->b6->a4->b2->d1->f2->h1->g3->
h5->g7->e8->c7->a6->b8->d7->f8->
h7->g5->h3->g1->e2->c1->a2->b4->
c2->a1->b3->a5->b7->d8->f7->h8->
g6->h4->g2->e1->f3->h2->f1->d2->
b1->a3->b5->a7->c8->e7->g8->h6->
g4->e3->f5->d6->c4->e5->c6->d4->
e6->c5->d3->f4->d5->f6->e4->c3
Knight's Tour from h8
h8->g6->h4->g2->e1->c2->a1->b3->
a5->b7->d8->f7->h6->g8->e7->c8->
a7->b5->a3->b1->d2->f1->h2->g4->
f2->h1->g3->h5->g7->e8->c7->a8->
b6->a4->b2->d1->c3->a2->c1->e2->
g1->h3->g5->h7->f8->d7->b8->a6->
b4->d3->c5->e6->f4->d5->f6->e4->
d6->f5->e3->c4->e5->c6->d4->f3

## Mathematica/Wolfram Language

Solution

knightsTourMoves[start_] :=
Module[{
vertexLabels = (# -> [email protected][[Quotient[# - 1, 8] + 1]] <> ToString[Mod[# - 1, 8] + 1]) & /@ Range[64], knightsGraph,
hamiltonianCycle, end},
knightsGraph = KnightTourGraph[i, i, VertexLabels -> vertexLabels, ImagePadding -> 15];
hamiltonianCycle = ((FindHamiltonianCycle[knightsGraph] /. UndirectedEdge -> DirectedEdge) /. labels)[[1]];
end = Cases[hamiltonianCycle, (x_ \[DirectedEdge] start) :> x][[1]];
FindShortestPath[g, start, end]]

Usage

knightsTourMoves["d8"]

(* out *)
{"d8", "e6", "d4", "c2", "a1", "b3", "a5", "b7", "c5", "a4", "b2", "c4", "a3", "b1", "c3", "a2", "b4", "a6", "b8", "c6", "a7", "b5", \
"c7", "a8", "b6", "c8", "d6", "e4", "d2", "f1", "e3", "d1", "f2", "h1", "g3", "e2", "c1", "d3", "e1", "g2", "h4", "f5", "e7", "d5", \
"f4", "h5", "g7", "e8", "f6", "g8", "h6", "g4", "h2", "f3", "g1", "h3", "g5", "h7", "f8", "d7", "e5", "g6", "h8", "f7"}

Analysis

vertexLabels replaces the default vertex (i.e. square) names of the chessboard with the standard algebraic names "a1", "a2",...,"h8".

vertexLabels = (# -> [email protected][[Quotient[# - 1, 8] + 1]] <> ToString[Mod[# - 1, 8] + 1]) & /@ Range[64]

(* out *)
{1 -> "a1", 2 -> "a2", 3 -> "a3", 4 -> "a4", 5 -> "a5", 6 -> "a6", 7 -> "a7", 8 -> "a8",
9 -> "b1", 10 -> "b2", 11 -> "b3", 12 -> "b4", 13 -> "b5", 14 -> "b6", 15 -> "b7", 16 -> "b8",
17 -> "c1", 18 -> "c2", 19 -> "c3", 20 -> "c4", 21 -> "c5", 22 -> "c6", 23 -> "c7", 24 -> "c8",
25 -> "d1", 26 -> "d2", 27 -> "d3", 28 -> "d4", 29 -> "d5", 30 -> "d6", 31 -> "d7", 32 -> "d8",
33 -> "e1", 34 -> "e2", 35 -> "e3", 36 -> "e4", 37 -> "e5", 38 -> "e6", 39 -> "e7", 40 -> "e8",
41 -> "f1", 42 -> "f2", 43 -> "f3", 44 -> "f4", 45 -> "f5", 46 -> "f6", 47 -> "f7", 48 -> "f8",
49 -> "g1", 50 -> "g2", 51 -> "g3", 52 -> "g4", 53 -> "g5", 54 -> "g6",55 -> "g7", 56 -> "g8",
57 -> "h1", 58 -> "h2", 59 -> "h3", 60 -> "h4", 61 -> "h5", 62 -> "h6", 63 -> "h7", 64 -> "h8"}

knightsGraph creates a graph of the solution space.

knightsGraph = KnightTourGraph[i, i, VertexLabels -> vertexLabels,  ImagePadding -> 15];

Find a Hamiltonian cycle (a path that visits each square exactly one time.)

hamiltonianCycle = ((FindHamiltonianCycle[knightsGraph] /. UndirectedEdge -> DirectedEdge) /. labels)[[1]];

Find the end square:

end = Cases[hamiltonianCycle, (x_ \[DirectedEdge] start) :> x][[1]];

Find shortest path from the start square to the end square.

FindShortestPath[g, start, end]]

## Mathprog

While a little slower than using Warnsdorff this solution is interesting:

1. It shows that Hidato and Knights Tour are essentially the same problem.

2. It is possible to specify which square is used for any Knights Move.

/*Knights.mathprog

Find a Knights Tour

Nigel_Galloway
January 11th., 2012
*/

param ZBLS;
param ROWS;
param COLS;
param D := 2;
set ROWSR := 1..ROWS;
set COLSR := 1..COLS;
set ROWSV := (1-D)..(ROWS+D);
set COLSV := (1-D)..(COLS+D);
param Iz{ROWSR,COLSR}, integer, default 0;
set ZBLSV := 1..(ZBLS+1);
set ZBLSR := 1..ZBLS;

var BR{ROWSV,COLSV,ZBLSV}, binary;

void0{r in ROWSV, z in ZBLSR,c in (1-D)..0}: BR[r,c,z] = 0;
void1{r in ROWSV, z in ZBLSR,c in (COLS+1)..(COLS+D)}: BR[r,c,z] = 0;
void2{c in COLSV, z in ZBLSR,r in (1-D)..0}: BR[r,c,z] = 0;
void3{c in COLSV, z in ZBLSR,r in (ROWS+1)..(ROWS+D)}: BR[r,c,z] = 0;
void4{r in ROWSV,c in (1-D)..0}: BR[r,c,ZBLS+1] = 1;
void5{r in ROWSV,c in (COLS+1)..(COLS+D)}: BR[r,c,ZBLS+1] = 1;
void6{c in COLSV,r in (1-D)..0}: BR[r,c,ZBLS+1] = 1;
void7{c in COLSV,r in (ROWS+1)..(ROWS+D)}: BR[r,c,ZBLS+1] = 1;

Izfree{r in ROWSR, c in COLSR, z in ZBLSR : Iz[r,c] = -1}: BR[r,c,z] = 0;
Iz1{Izr in ROWSR, Izc in COLSR, r in ROWSR, c in COLSR, z in ZBLSR : Izr=r and Izc=c and Iz[Izr,Izc]=z}: BR[r,c,z] = 1;

rule1{z in ZBLSR}: sum{r in ROWSR, c in COLSR} BR[r,c,z] = 1;
rule2{r in ROWSR, c in COLSR}: sum{z in ZBLSV} BR[r,c,z] = 1;
rule3{r in ROWSR, c in COLSR, z in ZBLSR}: BR[0,0,z+1] + BR[r-1,c-2,z+1] + BR[r-1,c+2,z+1] + BR[r-2,c-1,z+1] + BR[r-2,c+1,z+1] + BR[r+1,c+2,z+1] + BR[r+1,c-2,z+1] + BR[r+2,c-1,z+1] + BR[r+2,c+1,z+1] - BR[r,c,z] >= 0;

solve;

for {r in ROWSR} {
for {c in COLSR} {
printf " %2d", sum{z in ZBLSR} BR[r,c,z]*z;
}
printf "\n";
}
data;

param ROWS := 5;
param COLS := 5;
param ZBLS := 25;
param
Iz: 1 2 3 4 5 :=
1 . . . . .
2 . 19 2 . .
3 . . . . .
4 . . . . .
5 . . . . .
;

end;

Produces:

GLPSOL: GLPK LP/MIP Solver, v4.47
Parameter(s) specified in the command line:
--minisat --math Knights.mathprog
Generating void0...
Generating void1...
Generating void2...
Generating void3...
Generating void4...
Generating void5...
Generating void6...
Generating void7...
Generating Izfree...
Generating Iz1...
Generating rule1...
Generating rule2...
Generating rule3...
Model has been successfully generated
Will search for ANY feasible solution
Translating to CNF-SAT...
Original problem has 2549 rows, 2106 columns, and 9349 non-zeros
575 covering inequalities
1924 partitioning equalities
Solving CNF-SAT problem...
Instance has 3356 variables, 10874 clauses, and 34549 literals
==================================[MINISAT]===================================
| Conflicts | ORIGINAL | LEARNT | Progress |
| | Clauses Literals | Limit Clauses Literals Lit/Cl | |
==============================================================================
| 0 | 9000 32675 | 3000 0 0 0.0 | 0.000 % |
| 101 | 6025 21551 | 3300 93 1620 17.4 | 57.688 % |
| 251 | 6025 21551 | 3630 243 4961 20.4 | 57.688 % |
==============================================================================
SATISFIABLE
Objective value = 0.000000000e+000
Time used: 0.0 secs
Memory used: 6.5 Mb (6775701 bytes)
1 12 7 18 3
6 19 2 13 8
11 22 15 4 17
20 5 24 9 14
23 10 21 16 25
Model has been successfully processed

and

/*Knights.mathprog

Find a Knights Tour

Nigel_Galloway
January 11th., 2012
*/

param ZBLS;
param ROWS;
param COLS;
param D := 2;
set ROWSR := 1..ROWS;
set COLSR := 1..COLS;
set ROWSV := (1-D)..(ROWS+D);
set COLSV := (1-D)..(COLS+D);
param Iz{ROWSR,COLSR}, integer, default 0;
set ZBLSV := 1..(ZBLS+1);
set ZBLSR := 1..ZBLS;

var BR{ROWSV,COLSV,ZBLSV}, binary;

void0{r in ROWSV, z in ZBLSR,c in (1-D)..0}: BR[r,c,z] = 0;
void1{r in ROWSV, z in ZBLSR,c in (COLS+1)..(COLS+D)}: BR[r,c,z] = 0;
void2{c in COLSV, z in ZBLSR,r in (1-D)..0}: BR[r,c,z] = 0;
void3{c in COLSV, z in ZBLSR,r in (ROWS+1)..(ROWS+D)}: BR[r,c,z] = 0;
void4{r in ROWSV,c in (1-D)..0}: BR[r,c,ZBLS+1] = 1;
void5{r in ROWSV,c in (COLS+1)..(COLS+D)}: BR[r,c,ZBLS+1] = 1;
void6{c in COLSV,r in (1-D)..0}: BR[r,c,ZBLS+1] = 1;
void7{c in COLSV,r in (ROWS+1)..(ROWS+D)}: BR[r,c,ZBLS+1] = 1;

Izfree{r in ROWSR, c in COLSR, z in ZBLSR : Iz[r,c] = -1}: BR[r,c,z] = 0;
Iz1{Izr in ROWSR, Izc in COLSR, r in ROWSR, c in COLSR, z in ZBLSR : Izr=r and Izc=c and Iz[Izr,Izc]=z}: BR[r,c,z] = 1;

rule1{z in ZBLSR}: sum{r in ROWSR, c in COLSR} BR[r,c,z] = 1;
rule2{r in ROWSR, c in COLSR}: sum{z in ZBLSV} BR[r,c,z] = 1;
rule3{r in ROWSR, c in COLSR, z in ZBLSR}: BR[0,0,z+1] + BR[r-1,c-2,z+1] + BR[r-1,c+2,z+1] + BR[r-2,c-1,z+1] + BR[r-2,c+1,z+1] + BR[r+1,c+2,z+1] + BR[r+1,c-2,z+1] + BR[r+2,c-1,z+1] + BR[r+2,c+1,z+1] - BR[r,c,z] >= 0;

solve;

for {r in ROWSR} {
for {c in COLSR} {
printf " %2d", sum{z in ZBLSR} BR[r,c,z]*z;
}
printf "\n";
}
data;

param ROWS := 8;
param COLS := 8;
param ZBLS := 64;
param
Iz: 1 2 3 4 5 6 7 8 :=
1 . . . . . . . .
2 . . . . . . 48 .
3 . . . . . . . .
4 . . . . . . . .
5 . . . . . . . .
6 . . . . . . . .
7 . 58 . . . . . .
8 . . . . . . . .
;

end;

Produces:

GLPSOL: GLPK LP/MIP Solver, v4.47
Parameter(s) specified in the command line:
--minisat --math Knights.mathprog
Generating void0...
Generating void1...
Generating void2...
Generating void3...
Generating void4...
Generating void5...
Generating void6...
Generating void7...
Generating Izfree...
Generating Iz1...
Generating rule1...
Generating rule2...
Generating rule3...
Model has been successfully generated
Will search for ANY feasible solution
Translating to CNF-SAT...
Original problem has 10466 rows, 9360 columns, and 55330 non-zeros
3968 covering inequalities
6370 partitioning equalities
Solving CNF-SAT problem...
Instance has 15056 variables, 46754 clauses, and 149794 literals
==================================[MINISAT]===================================
| Conflicts | ORIGINAL | LEARNT | Progress |
| | Clauses Literals | Limit Clauses Literals Lit/Cl | |
==============================================================================
| 0 | 40512 143552 | 13504 0 0 0.0 | 0.000 % |
| 100 | 32458 114610 | 14854 89 5138 57.7 | 46.633 % |
| 250 | 32458 114610 | 16340 239 18544 77.6 | 46.633 % |
| 475 | 27499 102956 | 17974 424 42212 99.6 | 46.892 % |
| 813 | 27366 102490 | 19771 757 73184 96.7 | 51.541 % |
| 1322 | 27366 102490 | 21748 1264 137991 109.2 | 52.245 % |
| 2083 | 23226 92730 | 23923 2010 250286 124.5 | 53.620 % |
| 3227 | 22239 90284 | 26315 3138 460582 146.8 | 53.620 % |
| 4937 | 22239 90284 | 28947 4848 769486 158.7 | 53.620 % |
| 7499 | 22206 90168 | 31842 7404 1258240 169.9 | 55.167 % |
| 11346 | 21067 87284 | 35026 11248 2085553 185.4 | 55.167 % |
| 17113 | 21067 87284 | 38528 17015 3625910 213.1 | 55.167 % |
| 25763 | 21067 87284 | 42381 25665 5906283 230.1 | 55.167 % |
| 38738 | 21051 87252 | 46619 38638 9316878 241.1 | 55.679 % |
| 58199 | 21051 87252 | 51281 16434 3967196 241.4 | 55.685 % |
| 87393 | 20707 86474 | 56410 45624 13013357 285.2 | 56.277 % |
| 131184 | 20180 84834 | 62051 37252 8996727 241.5 | 56.542 % |
| 196871 | 20180 84834 | 68256 49392 13807861 279.6 | 56.542 % |
| 295399 | 20180 84834 | 75081 22688 5827696 256.9 | 56.542 % |
==============================================================================
SATISFIABLE
Objective value = 0.000000000e+000
Time used: 333.0 secs
Memory used: 28.2 Mb (29609617 bytes)
51 24 31 6 49 26 33 64
30 5 50 25 32 63 48 43
23 52 7 4 27 44 15 34
8 29 60 45 62 47 42 17
59 22 53 28 3 16 35 14
54 9 56 61 46 39 18 41
21 58 11 38 19 2 13 36
10 55 20 57 12 37 40 1
Model has been successfully processed

## Nim

Translation of: C++

This is a translation of the C++ and D versions with some changes, the most important being the addition of a check to detect that there is no solution. Without this check, the program crashes with an IndexError as Nim in debug and in release modes generates code to insure that indexes are valid.

We have added a case to test the absence of solution. Note that, in this case, there is a lot of backtracking which considerably slows down the execution.

import algorithm, options, random, parseutils, strutils, strformat

type
Board[N: static Positive] = array[N, array[N, int]]
Move = tuple[x, y: int]
MoveList = array[8, Move]
MoveIndexes = array[8, int]

const Moves: MoveList = [(2, 1), (1, 2), (-1, 2), (-2, 1), (-2, -1), (-1, -2), (1, -2), (2, -1)]

proc `\$`(board: Board): string =
## Display the board.
let size = len(\$(board.N * board.N)) + 1
for row in board:
for val in row:
stdout.write (\$val).align(size)
echo ""

proc sortedMoves(board: Board; x, y: int): MoveIndexes =
## Return the list of moves sorted by count of possible moves.

var counts: array[8, tuple[value, index: int]]
for i, d1 in Moves:
var count = 0
for d2 in Moves:
let x2 = x + d1.x + d2.x
let y2 = y + d1.y + d2.y
if x2 in 0..<board.N and y2 in 0..<board.N and board[y2][x2] == 0:
inc count
counts[i] = (count, i)

counts.shuffle() # Shuffle to randomly break ties.
counts.sort() # Lexicographic sort.

for i, count in counts:
result[i] = count.index

proc knightTour[N: static Positive](start: string): Option[Board[N]] =
## Return the knight tour for a board of size N x N and the starting
## position "start.
## If no solution is found, return "node" else return "some".

# Initialize the board with the starting position.
var board: Board[N]
var startx, starty: int
startx = ord(start[0]) - ord('a')
if startx notin 0..<N:
raise newException(ValueError, "wrong column.")
if parseInt(start, starty, 1) != start.len - 1 or starty notin 1..N:
raise newException(ValueError, "wrong line.")
starty = N - starty
board[starty][startx] = 1

type OrderItem = tuple[x, y, idx: int; mi: MoveIndexes]
var order: array[N * N, OrderItem]
order[0] = (startx, starty, 0, board.sortedMoves(startx, starty))

# Search a tour.
var n = 0
while n < N * N - 1:
let x = order[n].x
let y = order[n].y
var ok = false

for i in order[n].idx..7:
let d = Moves[order[n].mi[i]]
if x + d.x notin 0..<N or y + d.y notin 0..<N: continue
if board[y + d.y][x + d.x] == 0:
order[n].idx = i + 1
inc n
board[y + d.y][x + d.x] = n + 1
order[n] = (x + d.x, y + d.y, 0, board.sortedMoves(x + d.x, y + d.y))
ok = true
break

if not ok:
# Failed: backtrack.
echo "backtrack"
board[y][x] = 0
dec n
if n < 0: return none(Board[N]) # No solution found.

result = some(board)

proc run[N: static Positive](start: string) =
## Run the algorithm and display the result.
let result = knightTour[N](start)
echo &"Board size: {N}x{N}, starting position: {start}."
if result.isSome(): echo result.get()
else: echo "No solution found.\n"

when isMainModule:

randomize()

run[5]("c3")
#run[5]("c4") # No solution, so very slow compared to other cases.
run[8]("b5")
run[31]("a1")
Output:
Board size: 5x5, starting position: c3.
23 16 11  6 21
10  5 22 17 12
15 24  1 20  7
4  9 18 13  2
25 14  3  8 19

Board size: 5x5, starting position: c4.
No solution found.

Board size: 8x8, starting position: b5.
63 20  3 24 59 36  5 26
2 23 64 37  4 25 58 35
19 62 21 50 55 60 27  6
22  1 54 61 38 45 34 57
53 18 49 44 51 56  7 28
12 15 52 39 46 31 42 33
17 48 13 10 43 40 29  8
14 11 16 47 30  9 32 41

Board size: 31x31, starting position: a1.
275 112  19 116 277 604  21 118 823 770  23 120 961 940  25 122 943 926  27 124 917 898  29 126 911 872  31 128 197 870  33
18 115 276 601  20 117 772 767  22 119 958 851  24 121 954 941  26 123 936 925  28 125 912 899  30 127 910 871  32 129 198
111 274 113 278 605 760 603 822 771 824 769 948 957 960 939 944 953 942 927 916 929 918 897 908 913 900 873 196 875  34 869
114  17 600 273 602 775 766 773 768 949 850 959 852 947 952 955 932 937 930 935 924 915 920 905 894 909 882 901 868 199 130
271 110 279 606 759 610 761 776 821 764 825 816 951 956 853 938 945 934 923 928 919 896 893 914 907 904 867 874 195 876  35
16 581 272 599 280 607 774 765 762 779 950 849 826 815 946 933 854 931 844 857 890 921 906 895 886 883 902 881 200 131 194
109 270 281 580 609 758 611 744 777 820 763 780 817 848 827 808 811 846 855 922 843 858 889 892 903 866 885 192 877  36 201
282  15 582 269 598 579 608 757 688 745 778 819 754 783 814 847 828 807 810 845 856 891 842 859 884 887 880 863 202 193 132
267 108 283 578 583 612 689 614 743 756 691 746 781 818 753 784 809 812 829 806 801 840 835 888 865 862 203 878 191 530  37
14 569 268 585 284 597 576 619 690 687 742 755 692 747 782 813 752 785 802 793 830 805 860 841 836 879 864 529 204 133 190
107 266 285 570 577 584 613 686 615 620 695 684 741 732 711 748 739 794 751 786 803 800 839 834 861 528 837 188 531  38 205
286  13 568 265 586 575 596 591 618 685 616 655 696 693 740 733 712 749 738 795 792 831 804 799 838 833 722 527 206 189 134
263 106 287 508 571 590 587 574 621 592 639 694 683 656 731 710 715 734 787 750 737 796 791 832 721 798 207 532 187 474  39
12 417 264 567 288 509 572 595 588 617 654 657 640 697 680 713 730 709 716 735 788 727 720 797 790 723 526 473 208 135 186
105 262 289 416 507 566 589 512 573 622 593 638 653 682 659 698 679 714 729 708 717 736 789 726 719 472 533 184 475  40 209
290  11 418 261 502 415 510 565 594 513 562 641 658 637 652 681 660 699 678 669 728 707 718 675 724 525 704 471 210 185 136
259 104 291 414 419 506 503 514 511 564 623 548 561 642 551 636 651 670 661 700 677 674 725 706 703 534 211 476 183 396  41
10 331 260 493 292 501 420 495 504 515 498 563 624 549 560 643 662 635 650 671 668 701 676 673 524 705 470 395 212 137 182
103 258 293 330 413 494 505 500 455 496 547 516 485 552 625 550 559 644 663 634 649 672 667 702 535 394 477 180 397  42 213
294   9 332 257 492 329 456 421 490 499 458 497 546 517 484 553 626 543 558 645 664 633 648 523 666 469 536 393 220 181 138
255 102 295 328 333 412 491 438 457 454 489 440 459 486 545 518 483 554 627 542 557 646 665 632 537 478 221 398 179 214  43
8 319 256 335 296 345 326 409 422 439 436 453 488 441 460 451 544 519 482 555 628 541 522 647 468 631 392 219 222 139 178
101 254 297 320 327 334 411 346 437 408 423 368 435 452 487 442 461 450 445 520 481 556 629 538 479 466 399 176 215  44 165
298   7 318 253 336 325 344 349 410 347 360 407 424 383 434 427 446 443 462 449 540 521 480 467 630 391 218 223 164 177 140
251 100 303 300 321 316 337 324 343 350 369 382 367 406 425 384 433 428 447 444 463 430 539 390 465 400 175 216 169 166  45
6 299 252 317 304 301 322 315 348 361 342 359 370 381 366 405 426 385 432 429 448 389 464 401 174 217 224 163 150 141 168
99 250 241 302 235 248 307 338 323 314 351 362 341 358 371 380 365 404 377 386 431 402 173 388 225 160 153 170 167  46 143
240   5  98 249 242 305 234 247 308 339 232 313 352 363 230 357 372 379 228 403 376 387 226 159 154 171 162 149 142 151  82
63   2 239  66  97 236 243 306 233 246 309 340 231 312 353 364 229 356 373 378 227 158 375 172 161 148 155 152  83 144  47
4  67  64  61 238  69  96  59 244  71  94  57 310  73  92  55 354  75  90  53 374  77  88  51 156  79  86  49 146  81  84
1  62   3  68  65  60 237  70  95  58 245  72  93  56 311  74  91  54 355  76  89  52 157  78  87  50 147  80  85  48 145

## Perl

Knight's tour using Warnsdorffs algorithm

use strict;
use warnings;
# Find a knight's tour

my @board;

# Choose starting position - may be passed in on command line; if
# not, choose random square.
my (\$i, \$j);
if (my \$sq = shift @ARGV) {
die "\$0: illegal start square '\$sq'\n" unless (\$i, \$j) = from_algebraic(\$sq);
} else {
(\$i, \$j) = (int rand 8, int rand 8);
}

# Move sequence
my @moves = ();

foreach my \$move (1..64) {
# Record current move
push @moves, to_algebraic(\$i,\$j);
\$board[\$i][\$j] = \$move;

# Get list of possible next moves
my @targets = possible_moves(\$i,\$j);

# Find the one with the smallest degree
my @min = (9);
foreach my \$target (@targets) {
my (\$ni, \$nj) = @\$target;
my \$next = possible_moves(\$ni,\$nj);
@min = (\$next, \$ni, \$nj) if \$next < \$min[0];
}

# And make it
(\$i, \$j) = @min[1,2];
}

# Print the move list
for (my \$i=0; \$i<4; ++\$i) {
for (my \$j=0; \$j<16; ++\$j) {
my \$n = \$i*16+\$j;
print \$moves[\$n];
print ', ' unless \$n+1 >= @moves;
}
print "\n";
}
print "\n";

# And the board, with move numbers
for (my \$i=0; \$i<8; ++\$i) {
for (my \$j=0; \$j<8; ++\$j) {
# Assumes (1) ANSI sequences work, and (2) output
# is light text on a dark background.
print "\e[7m" if (\$i%2==\$j%2);
printf " %2d", \$board[\$i][\$j];
print "\e[0m";
}
print "\n";
}

# Find the list of positions the knight can move to from the given square
sub possible_moves
{
my (\$i, \$j) = @_;
return grep { \$_->[0] >= 0 && \$_->[0] < 8
&& \$_->[1] >= 0 && \$_->[1] < 8
&& !\$board[\$_->[0]][\$_->[1]] } (
[\$i-2,\$j-1], [\$i-2,\$j+1], [\$i-1,\$j-2], [\$i-1,\$j+2],
[\$i+1,\$j-2], [\$i+1,\$j+2], [\$i+2,\$j-1], [\$i+2,\$j+1]);
}

# Return the algebraic name of the square identified by the coordinates
# i=rank, 0=black's home row; j=file, 0=white's queen's rook
sub to_algebraic
{
my (\$i, \$j) = @_;
chr(ord('a') + \$j) . (8-\$i);
}

# Return the coordinates matching the given algebraic name
sub from_algebraic
{
my \$square = shift;
return unless \$square =~ /^([a-h])([1-8])\$/;
return (8-\$2, ord(\$1) - ord('a'));
}

Sample output (start square c3):

## Phix

This is pretty fast (<<1s) up to size 48, before some sizes start to take quite some time to complete. It will even solve a 200x200 in 0.67s

constant size = 8
constant nchars = length(sprintf(" %d",size*size))
constant fmt = sprintf(" %%%dd",nchars-1)
constant blank = repeat(' ',nchars)

-- to simplify output, each square is nchars
sequence board = repeat(repeat(' ',size*nchars),size)
-- keep current counts, immediately backtrack if any hit 0
-- (in line with the above, we only use every nth entry)
sequence warnsdorffs = repeat(repeat(0,size*nchars),size)

constant ROW = 1, COL = 2
constant moves = {{-1,-2},{-2,-1},{-2,1},{-1,2},{1,2},{2,1},{2,-1},{1,-2}}

function onboard(integer row, integer col)
return row>=1 and row<=size and col>=nchars and col<=nchars*size
end function

procedure init_warnsdorffs()
integer nrow,ncol
for row=1 to size do
for col=nchars to nchars*size by nchars do
for move=1 to length(moves) do
nrow = row+moves[move][ROW]
ncol = col+moves[move][COL]*nchars
if onboard(nrow,ncol) then
warnsdorffs[row][col] += 1
end if
end for
end for
end for
end procedure

atom t0 = time()
integer tries = 0
atom t1 = time()+1
function solve(integer row, integer col, integer n)
integer nrow, ncol
if time()>t1 then
?{row,floor(col/nchars),n,tries}
puts(1,join(board,"\n"))
t1 = time()+1
-- if wait_key()='!' then ?9/0 end if
end if
tries+= 1
if n>size*size then return 1 end if
sequence wmoves = {}
for move=1 to length(moves) do
nrow = row+moves[move][ROW]
ncol = col+moves[move][COL]*nchars
if onboard(nrow,ncol)
and board[nrow][ncol]=' ' then
wmoves = append(wmoves,{warnsdorffs[nrow][ncol],nrow,ncol})
end if
end for
wmoves = sort(wmoves)
-- avoid creating orphans
if length(wmoves)<2 or wmoves[2][1]>1 then
for m=1 to length(wmoves) do
{?,nrow,ncol} = wmoves[m]
warnsdorffs[nrow][ncol] -= 1
end for
for m=1 to length(wmoves) do
{?,nrow,ncol} = wmoves[m]
board[nrow][ncol-nchars+1..ncol] = sprintf(fmt,n)
if solve(nrow,ncol,n+1) then return 1 end if
board[nrow][ncol-nchars+1..ncol] = blank
end for
for m=1 to length(wmoves) do
{?,nrow,ncol} = wmoves[m]
warnsdorffs[nrow][ncol] += 1
end for
end if
return 0
end function

init_warnsdorffs()
board[1][nchars] = '1'
if solve(1,nchars,2) then
puts(1,join(board,"\n"))
printf(1,"\nsolution found in %d tries (%3.2fs)\n",{tries,time()-t0})
else
puts(1,"no solutions found\n")
end if

{} = wait_key()
Output:
"started"
1 16 31 40  3 18 21 56
30 39  2 17 42 55  4 19
15 32 41 46 53 20 57 22
38 29 48 43 58 45 54  5
33 14 37 52 47 60 23 62
28 49 34 59 44 63  6  9
13 36 51 26 11  8 61 24
50 27 12 35 64 25 10  7
solution found in 64 tries (0.00s)

## PicoLisp

# Build board
(grid 8 8)

# Generate legal moves for a given position
(de moves (Tour)
(extract
'((Jump)
(let? Pos (Jump (car Tour))
(unless (memq Pos Tour)
Pos ) ) )
(quote # (taken from "games/chess.l")
((This) (: 0 1 1 0 -1 1 0 -1 1)) # South Southwest
((This) (: 0 1 1 0 -1 1 0 1 1)) # West Southwest
((This) (: 0 1 1 0 -1 -1 0 1 1)) # West Northwest
((This) (: 0 1 1 0 -1 -1 0 -1 -1)) # North Northwest
((This) (: 0 1 -1 0 -1 -1 0 -1 -1)) # North Northeast
((This) (: 0 1 -1 0 -1 -1 0 1 -1)) # East Northeast
((This) (: 0 1 -1 0 -1 1 0 1 -1)) # East Southeast
((This) (: 0 1 -1 0 -1 1 0 -1 1)) ) ) ) # South Southeast

# Build a list of moves, using Warnsdorff’s algorithm
(let Tour '(b1) # Start at b1
(while
(mini
'((P) (length (moves (cons P Tour))))
(moves Tour) )
(push 'Tour @) )
(flip Tour) )

Output:

-> (b1 a3 b5 a7 c8 b6 a8 c7 a6 b8 d7 f8 h7 g5 h3 g1 e2 c1 a2 b4 c2 a1 b3 a5 b7
d8 c6 d4 e6 c5 a4 c3 d1 b2 c4 d2 f1 h2 f3 e1 d3 e5 f7 h8 g6 h4 g2 f4 d5 e7 g8
h6 g4 e3 f5 d6 e8 g7 h5 f6 e4 g3 h1 f2)

## PostScript

You probably shouldn't send this to a printer. Solution using Warnsdorffs algorithm.

%%BoundingBox: 0 0 300 300

/s { 300 n div } def
/l { rlineto } def

% draws a square
/bx { s mul exch s mul moveto s 0 l 0 s l s neg 0 l 0 s neg l } def

% draws checker board
/xbd { 1 setgray
0 0 moveto 300 0 l 0 300 l -300 0 l fill
.7 1 .6 setrgbcolor
0 1 n1 { dup 2 mod 2 n1 { 1 index bx fill } for pop } for
0 setgray
} def

/ar1 { [ exch { 0 } repeat ] } def
/ar2 { [ exch dup { dup ar1 exch } repeat pop ] } def

/neighbors {
-1 2 0
1 2 0
2 1 0
2 -1 0
1 -2 0
-1 -2 0
-2 -1 0
-2 1 0
%24 x y add 3 mul roll
} def

/func { 0 dict begin mark } def
/var { counttomark -1 1 { 2 add -1 roll def } for cleartomark } def

% x y can_goto -> bool
/can_goto {
func /x /y var
x 0 ge
x n lt
y 0 ge
y n lt
and and and {
occupied x get y get 0 eq
} { false } ifelse
end
} def

% x y num_access -> number of cells reachable from (x,y)
/num_access {
func /x /y var
/count 0 def
x y can_goto {
neighbors
} if
} repeat
count 0 gt { count } { 9 } ifelse
} { 10 } ifelse
end
} def

% a circle
/marker { x s mul y s mul s 20 div 0 360 arc fill } def

% n solve -> draws board of size n x n, calcs path and draws it
/solve {
func /n var
/n1 n 1 sub def

/c false def

8 n div setlinewidth
gsave

0 1 n1 { /x exch def c not {
0 1 n1 {
/occupied n ar2 def
c not {
/c true def
/y exch def
grestore xbd gsave
s 2 div dup translate
n n mul 2 sub -1 0 { /iter exch def
c {
0 setgray marker x s mul y s mul moveto
occupied x get y 1 put
neighbors
8 { pop y add exch x add exch 2 copy num_access 24 3 roll } repeat
7 { dup 4 index lt { 6 3 roll } if pop pop pop } repeat

9 ge iter 0 gt and { /c false def } if
/y exch def
/x exch def
.2 setgray x s mul y s mul lineto stroke
} if } for
% to be nice, draw box at final position
.5 0 0 setrgbcolor marker
y .5 sub x .5 sub bx 1 setlinewidth stroke
stroke
} if
} for } if } for showpage
grestore
end
} def

3 1 100 { solve } for

%%EOF

## Prolog

Works with: SWI-Prolog

Knights tour using Warnsdorffs algorithm

% N is the number of lines of the chessboard
knight(N) :-
Max is N * N,
length(L, Max),
knight(N, 0, Max, 0, 0, L),
display(N, 0, L).

% knight(NbCol, Coup, Max, Lig, Col, L),
% NbCol : number of columns per line
% Coup  : number of the current move
% Max  : maximum number of moves
% Lig/ Col : current position of the knight
% L : the "chessboard"

% the game is over
knight(_, Max, Max, _, _, _) :- !.

knight(NbCol, N, MaxN, Lg, Cl, L) :-
% Is the move legal
Lg >= 0, Cl >= 0, Lg < NbCol, Cl < NbCol,

Pos is Lg * NbCol + Cl,
N1 is N+1,
% is the place free
nth0(Pos, L, N1),

LgM1 is Lg - 1, LgM2 is Lg - 2, LgP1 is Lg + 1, LgP2 is Lg + 2,
ClM1 is Cl - 1, ClM2 is Cl - 2, ClP1 is Cl + 1, ClP2 is Cl + 2,
maplist(best_move(NbCol, L),
[(LgP1, ClM2), (LgP2, ClM1), (LgP2, ClP1),(LgP1, ClP2),
(LgM1, ClM2), (LgM2, ClM1), (LgM2, ClP1),(LgM1, ClP2)],
R),
sort(R, RS),
pairs_values(RS, Moves),

move(NbCol, N1, MaxN, Moves, L).

move(NbCol, N1, MaxN, [(Lg, Cl) | R], L) :-
knight(NbCol, N1, MaxN, Lg, Cl, L);
move(NbCol, N1, MaxN, R, L).

%% An illegal move is scored 1000
best_move(NbCol, _L, (Lg, Cl), 1000-(Lg, Cl)) :-
( Lg < 0 ; Cl < 0; Lg >= NbCol; Cl >= NbCol), !.

best_move(NbCol, L, (Lg, Cl), 1000-(Lg, Cl)) :-
Pos is Lg*NbCol+Cl,
nth0(Pos, L, V),
\+var(V), !.

%% a legal move is scored with the number of moves a knight can make
best_move(NbCol, L, (Lg, Cl), R-(Lg, Cl)) :-
LgM1 is Lg - 1, LgM2 is Lg - 2, LgP1 is Lg + 1, LgP2 is Lg + 2,
ClM1 is Cl - 1, ClM2 is Cl - 2, ClP1 is Cl + 1, ClP2 is Cl + 2,
include(possible_move(NbCol, L),
[(LgP1, ClM2), (LgP2, ClM1), (LgP2, ClP1),(LgP1, ClP2),
(LgM1, ClM2), (LgM2, ClM1), (LgM2, ClP1),(LgM1, ClP2)],
Res),
length(Res, Len),
( Len = 0 -> R = 1000; R = Len).

% test if a place is enabled
possible_move(NbCol, L, (Lg, Cl)) :-
% move must be legal
Lg >= 0, Cl >= 0, Lg < NbCol, Cl < NbCol,
Pos is Lg * NbCol + Cl,
% place must be free
nth0(Pos, L, V),
var(V).

display(_, _, []).
display(N, N, L) :-
nl,
display(N, 0, L).

display(N, M, [H | T]) :-
writef('%3r', [H]),
M1 is M + 1,
display(N, M1, T).

Output :

?- knight(8).
1  16  31  40   3  18  21  56
30  39   2  17  42  55   4  19
15  32  41  46  53  20  57  22
38  29  48  43  58  45  54   5
33  14  37  52  47  60  23  62
28  49  34  59  44  63   6   9
13  36  51  26  11   8  61  24
50  27  12  35  64  25  10   7
true .

?- knight(20).
1  40  81  90   3  42  77  94   5  44  73 102   7  46  69  62   9  48  51  60
82  89   2  41  92  95   4  43  76 101   6  45  72 103   8  47  68  61  10  49
39  80  91  96 153  78  93 100 129  74 109 104 123  70 111 120  63  50  59  52
88  83 154  79  98 159 152  75 108 105 128  71 110 121 124  67 112 119  64  11
155  38  97 160 157 200  99 162 151 130 107 122 127 132 141 118 125  66  53  58
84  87 156 199 176 161 158 201 106 163 150 131 142 145 126 133 140 113  12  65
37 182  85 178 207 198 175 164 173 216 143 166 149 222 139 146 117 134  57  54
86 179 206 197 204 177 208 217 202 165 172 221 144 167 148 223 138  55 114  13
183  36 181 212 209 218 203 174 215 220 227 170 281 224 303 168 147 116 135  56
180 211 196 205 230 213 238 219 228 171 280 225 302 169 282 343 304 137  14 115
35 184 231 210 237 246 229 214 279 226 301 298 283 342 367 308 347 344 305 136
232 195 236 245 234 239 278 247 300 297 284 359 366 309 348 345 368 307 350  15
185  34 233 240 261 248 287 296 285 358 299 310 341 378 365 384 349 346 369 306
194 241 250 235 244 277 260 313 294 311 360 373 364 383 354 379 370 385  16 351
33 186 243 262 249 288 295 286 361 316 357 340 377 372 395 386 353 380 333 388
242 193 254 251 276 259 314 293 312 321 374 363 398 355 382 371 394 387 352  17
187  32 263 258 267 252 289 322 315 362 317 356 339 376 399 396 381 334 389 332
192 255 190 253 264 275 268 271 292 323 320 375 326 397 338 335 390 393  18  21
31 188 257 266  29 270 273 290  27 318 327 324  25 336 329 400  23  20 331 392
256 191  30 189 274 265  28 269 272 291  26 319 328 325  24 337 330 391  22  19
true .

### Alternative version

Works with: GNU Prolog
:- initialization(main).

board_size(8).
in_board(X*Y) :- board_size(N), between(1,N,Y), between(1,N,X).

% express jump-graph in dynamic "move"-rules
make_graph :-
findall(_, (in_board(P), assert_moves(P)), _).

% where
assert_moves(P) :-
findall(_, (can_move(P,Q), asserta(move(P,Q))), _).

can_move(X*Y,Q) :-
( one(X,X1), two(Y,Y1) ; two(X,X1), one(Y,Y1) )
, Q = X1*Y1, in_board(Q)
. % where
one(M,N) :- succ(M,N) ; succ(N,M).
two(M,N) :- N is M + 2 ; N is M - 2.

hamiltonian(P,Pn) :-
board_size(N), Size is N * N
, hamiltonian(P,Size,[],Ps), enumerate(Size,Ps,Pn)
.
% where
enumerate(_, [] , [] ).
enumerate(N, [P|Ps], [N:P|Pn]) :- succ(M,N), enumerate(M,Ps,Pn).

hamiltonian(P,N,Ps,Res) :-
N =:= 1 -> Res = [P|Ps]
; warnsdorff(Ps,P,Q), succ(M,N)
, hamiltonian(Q,M,[P|Ps],Res)
.
% where
warnsdorff(Ps,P,Q) :-
moves(Ps,P,Qs), maplist(next_moves(Ps), Qs, Xs)
, keysort(Xs,Ys), member(_-Q,Ys)
.
next_moves(Ps,Q,L-Q) :- moves(Ps,Q,Rs), length(Rs,L).

moves(Ps,P,Qs) :-
findall(Q, (move(P,Q), \+ member(Q,Ps)), Qs).

show_path(Pn) :- findall(_, (in_board(P), show_cell(Pn,P)), _).
% where
show_cell(Pn,X*Y) :-
member(N:X*Y,Pn), format('%3.0d',[N]), board_size(X), nl.

main :- make_graph, hamiltonian(5*3,Pn), show_path(Pn), halt.
Output:
5 18 35 22  3 16 55 24
36 21  4 17 54 23  2 15
19  6 59 34  1 14 25 56
60 37 20 53 62 57 32 13
7 52 61 58 33 30 63 26
38 49 40 29 64 45 12 31
41  8 51 48 43 10 27 46
50 39 42  9 28 47 44 1

## Python

Knights tour using Warnsdorffs algorithm

import copy

boardsize=6
_kmoves = ((2,1), (1,2), (-1,2), (-2,1), (-2,-1), (-1,-2), (1,-2), (2,-1))

def chess2index(chess, boardsize=boardsize):
'Convert Algebraic chess notation to internal index format'
chess = chess.strip().lower()
x = ord(chess[0]) - ord('a')
y = boardsize - int(chess[1:])
return (x, y)

def boardstring(board, boardsize=boardsize):
r = range(boardsize)
lines = ''
for y in r:
lines += '\n' + ','.join('%2i' % board[(x,y)] if board[(x,y)] else ' '
for x in r)
return lines

def knightmoves(board, P, boardsize=boardsize):
Px, Py = P
kmoves = set((Px+x, Py+y) for x,y in _kmoves)
kmoves = set( (x,y)
for x,y in kmoves
if 0 <= x < boardsize
and 0 <= y < boardsize
and not board[(x,y)] )
return kmoves

def accessibility(board, P, boardsize=boardsize):
access = []
brd = copy.deepcopy(board)
for pos in knightmoves(board, P, boardsize=boardsize):
brd[pos] = -1
access.append( (len(knightmoves(brd, pos, boardsize=boardsize)), pos) )
brd[pos] = 0
return access

def knights_tour(start, boardsize=boardsize, _debug=False):
board = {(x,y):0 for x in range(boardsize) for y in range(boardsize)}
move = 1
P = chess2index(start, boardsize)
board[P] = move
move += 1
if _debug:
print(boardstring(board, boardsize=boardsize))
while move <= len(board):
P = min(accessibility(board, P, boardsize))[1]
board[P] = move
move += 1
if _debug:
print(boardstring(board, boardsize=boardsize))
input('\n%2i next: ' % move)
return board

if __name__ == '__main__':
while 1:
boardsize = int(input('\nboardsize: '))
if boardsize < 5:
continue
start = input('Start position: ')
board = knights_tour(start, boardsize)
print(boardstring(board, boardsize=boardsize))
Sample runs
boardsize: 5
Start position: c3

19,12,17, 6,21
2, 7,20,11,16
13,18, 1,22, 5
8, 3,24,15,10
25,14, 9, 4,23

boardsize: 8
Start position: h8

38,41,18, 3,22,27,16, 1
19, 4,39,42,17, 2,23,26
40,37,54,21,52,25,28,15
5,20,43,56,59,30,51,24
36,55,58,53,44,63,14,29
9, 6,45,62,57,60,31,50
46,35, 8,11,48,33,64,13
7,10,47,34,61,12,49,32

boardsize: 10
Start position: e6

29, 4,57,24,73, 6,95,10,75, 8
58,23,28, 5,94,25,74, 7,100,11
3,30,65,56,27,72,99,96, 9,76
22,59, 2,63,68,93,26,81,12,97
31,64,55,66, 1,82,71,98,77,80
54,21,60,69,62,67,92,79,88,13
49,32,53,46,83,70,87,42,91,78
20,35,48,61,52,45,84,89,14,41
33,50,37,18,47,86,39,16,43,90
36,19,34,51,38,17,44,85,40,15

boardsize: 200
Start position: a1

510,499,502,101,508,515,504,103,506,5021 ... 195,8550,6691,6712,197,6704,201,6696,199
501,100,509,514,503,102,507,5020,5005,10 ... 690,6713,196,8553,6692,6695,198,6703,202
498,511,500,4989,516,5019,5004,505,5022, ... ,30180,8559,6694,6711,8554,6705,200,6697
99,518,513,4992,5003,4990,5017,5044,5033 ... 30205,8552,30181,8558,6693,6702,203,6706
512,497,4988,517,5018,5001,5034,5011,504 ... 182,30201,30204,8555,6710,8557,6698,6701
519,98,4993,5002,4991,5016,5043,5052,505 ... 03,30546,30183,30200,30185,6700,6707,204
496,4987,520,5015,5000,5035,5012,5047,51 ... 4,30213,30202,31455,8556,6709,30186,6699
97,522,4999,4994,5013,5042,5051,5060,505 ... 7,31456,31329,30184,30199,30190,205,6708
4986,495,5014,521,5036,4997,5048,5101,50 ... 1327,31454,30195,31472,30187,30198,30189
523,96,4995,4998,5041,5074,5061,5050,507 ... ,31330,31471,31328,31453,30196,30191,206

...

404,731,704,947,958,1013,966,1041,1078,1 ... 9969,39992,39987,39996,39867,39856,39859
5,706,735,960,955,972,957,1060,1025,106 ... ,39978,39939,39976,39861,39990,297,39866
724,403,730,705,946,967,1012,971,1040,10 ... 9975,39972,39991,39868,39863,39860,39855
707, 4,723,736,729,956,973,996,1061,1026 ... ,39850,39869,39862,39973,39852,39865,298
402,725,708,943,968,945,970,1011,978,997 ... 6567,39974,39851,39864,36571,39854,36573
3,722,737,728,741,942,977,974,995,1010, ... ,39800,39849,36570,39853,36574,299,14088
720,401,726,709,944,969,742,941,980,975, ... ,14091,36568,36575,14084,14089,36572,843
711, 2,721,738,727,740,715,976,745,940,9 ... 65,36576,14083,14090,36569,844,14087,300
400,719,710,713,398,717,746,743,396,981, ... ,849,304,14081,840,847,302,14085,842,845
1,712,399,718,739,714,397,716,747,744,3 ... 4078,839,848,303,14082,841,846,301,14086

The 200x200 example warmed my study in its computation but did return a tour.

P.S. There is a slight deviation to a strict interpretation of Warnsdorff's algorithm in that as a convenience, tuples of the length of the knight moves followed by the position are minimized so knights moves with the same length will try and break the ties based on their minimum x,y position. In practice, it seems to give comparable results to the original algorithm.

boardsize: 5 Start position: a3 Traceback (most recent call last):

File "rosettacodekt.py", line 65, in <module>
board = knights_tour(start, boardsize)
File "rosettacodekt.py", line 51, in knights_tour
P = min(accessibility(board, P, boardsize))[1]

ValueError: min() arg is an empty sequence

## R

Based on a slight modification of Warnsdorff's algorithm, in that if a dead-end is reached, the program backtracks to the next best move.

#!/usr/bin/Rscript

# M x N Chess Board.
M = 8; N = 8; board = matrix(0, nrow = M, ncol = N)

# Get/Set value on a board position.
getboard = function (position) { board[position[1], position[2]] }
setboard = function (position, x) { board[position[1], position[2]] <<- x }

# (Relative) Hops of a Knight.
hops = cbind(c(-2, -1), c(-1, -2), c(+1, -2), c(+2, -1),
c(+2, +1), c(+1, +2), c(-1, +2), c(-2, +1))

# Validate a move.
valid = function (move) {
all(1 <= move & move <= c(M, N)) && (getboard(move) == 0)
}

# Moves possible from a given position.
explore = function (position) {
moves = position + hops
cbind(moves[, apply(moves, 2, valid)])
}

# Possible moves sorted according to their Wornsdorff cost.
candidates = function (position) {
moves = explore(position)

# No candidate moves available.
if (ncol(moves) == 0) { return(moves) }

wcosts = apply(moves, 2, function (position) { ncol(explore(position)) })
cbind(moves[, order(wcosts)])
}

# Recursive function for touring the chess board.
knightTour = function (position, moveN) {

# Tour Complete.
if (moveN > (M * N)) {
print(board)
quit()
}

# Available moves.
moves = candidates(position)

# None possible. Backtrack.
if (ncol(moves) == 0) { return() }

# Make a move, and continue the tour.
apply(moves, 2, function (position) {
setboard(position, moveN)
knightTour(position, moveN + 1)
setboard(position, 0)
})
}

# User Input: Starting position (in algebraic notation).
square = commandArgs(trailingOnly = TRUE)

# Convert into board co-ordinates.
row = M + 1 - as.integer(substr(square, 2, 2))
ascii = function (ch) { as.integer(charToRaw(ch)) }
col = 1 + ascii(substr(square, 1, 1)) - ascii('a')
position = c(row, col)

# Begin tour.
setboard(position, 1); knightTour(position, 2)

Output:

./knight.R e3

[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
[1,]    6    9   24   55   62   11   26   29
[2,]   23   54    7   10   25   28   63   12
[3,]    8    5   50   61   56   59   30   27
[4,]   37   22   53   58   43   48   13   64
[5,]    4   51   38   49   60   57   44   31
[6,]   21   36   19   52    1   42   47   14
[7,]   18    3   34   39   16   45   32   41
[8,]   35   20   17    2   33   40   15   46

## Racket

#lang racket
(define N 8)
(define nexts ; construct the graph
(let ([ds (for*/list ([x 2] [x* '(+1 -1)] [y* '(+1 -1)])
(cons (* x* (+ 1 x)) (* y* (- 2 x))))])
(for*/vector ([i N] [j N])
(filter values (for/list ([d ds])
(let ([i (+ i (car d))] [j (+ j (cdr d))])
(and (< -1 i N) (< -1 j N) (+ j (* N i)))))))))
(define (tour x y)
(define xy (+ x (* N y)))
(let loop ([seen (list xy)] [ns (vector-ref nexts xy)] [n (sub1 (* N N))])
(if (zero? n) (reverse seen)
(for/or ([next (sort (map (λ(n) (cons n (remq* seen (vector-ref nexts n)))) ns)
< #:key length #:cache-keys? #t)])
(loop (cons (car next) seen) (cdr next) (sub1 n))))))
(define (draw tour)
(define v (make-vector (* N N)))
(for ([n tour] [i (in-naturals 1)]) (vector-set! v n i))
(for ([i N])
(displayln (string-join (for/list ([j (in-range i (* N N) N)])
(~a (vector-ref v j) #:width 2 #:align 'right))
" "))))
(draw (tour (random N) (random N)))

Output:
56 11 36 33 52 13 38 17
35 32 55 12 37 16 51 14
10 57 34 53 48 45 18 39
31 54 43 64 41 50 15 46
60  9 58 49 44 47 40 19
27 30 61 42 63 22  1  4
8 59 28 25  6  3 20 23
29 26  7 62 21 24  5  2

## Raku

(formerly Perl 6)

Translation of: Perl
Works with: rakudo version 2015-09-17
my @board;

my \$I = 8;
my \$J = 8;
my \$F = \$I*\$J > 99 ?? "%3d" !! "%2d";

# Choose starting position - may be passed in on command line; if
# not, choose random square.
my (\$i, \$j);

if my \$sq = shift @*ARGS {
die "\$*PROGRAM_NAME: illegal start square '\$sq'\n" unless (\$i, \$j) = from_algebraic(\$sq);
}
else {
(\$i, \$j) = (^\$I).pick, (^\$J).pick;
}

# Move sequence
my @moves = ();

for 1 .. \$I * \$J -> \$move {
# Record current move
push @moves, to_algebraic(\$i,\$j);
# @board[\$i] //= []; # (uncomment if autoviv is broken)
@board[\$i][\$j] = \$move;

# Find move with the smallest degree
my @min = (9);
for possible_moves(\$i,\$j) -> @target {
my (\$ni, \$nj) = @target;
my \$next = possible_moves(\$ni,\$nj);
@min = \$next, \$ni, \$nj if \$next < @min[0];
}

# And make it
(\$i, \$j) = @min[1,2];
}

# Print the move list
for @moves.kv -> \$i, \$m {
print ',', \$i %% 16 ?? "\n" !! " " if \$i;
print \$m;
}
say "\n";

# And the board, with move numbers
for ^\$I -> \$i {
for ^\$J -> \$j {
# Assumes (1) ANSI sequences work, and (2) output
# is light text on a dark background.
print "\e[7m" if \$i % 2 == \$j % 2;
printf \$F, @board[\$i][\$j];
print "\e[0m";
}
print "\n";
}

# Find the list of positions the knight can move to from the given square
sub possible_moves(\$i,\$j) {
grep -> [\$ni, \$nj] { \$ni ~~ ^\$I and \$nj ~~ ^\$J and !@board[\$ni][\$nj] },
[\$i-2,\$j-1], [\$i-2,\$j+1], [\$i-1,\$j-2], [\$i-1,\$j+2],
[\$i+1,\$j-2], [\$i+1,\$j+2], [\$i+2,\$j-1], [\$i+2,\$j+1];
}

# Return the algebraic name of the square identified by the coordinates
# i=rank, 0=black's home row; j=file, 0=white's queen's rook
sub to_algebraic(\$i,\$j) {
chr(ord('a') + \$j) ~ (\$I - \$i);
}

# Return the coordinates matching the given algebraic name
sub from_algebraic(\$square where /^ (<[a..z]>) (\d+) \$/) {
\$I - \$1, ord(~\$0) - ord('a');
}

(Output identical to Perl's above.)

## REXX

This REXX version is modeled after the XPL0 example.

The size of the chessboard may be specified as well as the knight's starting position.

This is an   open tour   solution.   (See this task's   discussion   page for an explanation, the section is   The 7x7 problem.)

/*REXX program solves the  knight's tour  problem   for a  (general)   NxN   chessboard.*/
parse arg N sRank sFile . /*obtain optional arguments from the CL*/
if N=='' | N=="," then N=8 /*No boardsize specified? Use default.*/
if sRank=='' | sRank=="," then sRank=N /*No starting rank given? " " */
if sFile=='' | sFile=="," then sFile=1 /* " " file " " " */
NN=N**2; NxN='a ' N"x"N ' chessboard' /*file [↓] [↓] r=rank */
@.=; do r=1 for N; do f=1 for N; @.r.f=.; end /*f*/; end /*r*/
beg= '-1-' /*[↑] create an empty NxN chessboard.*/
Kr = '2 1 -1 -2 -2 -1 1 2' /*the legal "rank" moves for a knight.*/
Kf = '1 2 2 1 -1 -2 -2 -1' /* " " "file" " " " " */
kr.M=words(Kr) /*number of possible moves for a Knight*/
parse var Kr Kr.1 Kr.2 Kr.3 Kr.4 Kr.5 Kr.6 Kr.7 Kr.8 /*parse the legal moves by hand*/
parse var Kf Kf.1 Kf.2 Kf.3 Kf.4 Kf.5 Kf.6 Kf.7 Kf.8 /* " " " " " " */
@.sRank.sFile= beg /*the knight's starting position. */
@kt= "knight's tour" /*a handy-dandy literal for the SAYs. */
if \move(2, sRank, sFile) & \(N==1) then say 'No' @kt "solution for" NxN'.'
else say 'A solution for the' @kt "on" NxN':'
!=left('', 9 * (n<18) ) /*used for indentation of chessboard. */
_=substr(copies("┼───",N),2); say; say  ! translate('┌'_"┐", '┬', "┼") /*a square.*/
/* [↓] build a display for chessboard.*/
do r=N for N by -1; if r\==N then say ! '├'_"┤"; [email protected].
do f=1 for N; [email protected].r.f; if ?==NN then ?='end'; L=L'│'center(?, 3) /*is "end"?*/
end /*f*/ /*done with rank of the chessboard.*/
say ! translate(L'│', , .) /*display a " " " " */
end /*r*/ /*19x19 chessboard can be shown 80 cols*/

say  ! translate('└'_"┘", '┴', "┼") /*show the last rank of the chessboard.*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
move: procedure expose @. Kr. Kf. NN; parse arg #,rank,file /*obtain move,rank,file.*/
do t=1 for Kr.M; nr=rank+Kr.t; nf=file+Kf.t /*position of the knight*/
if @.nr.nf==. then do; @.nr.nf=# /*Empty? Knight can move*/
if #==NN then return 1 /*is this the last move?*/
if move(#+1,nr,nf) then return 1 /* " " " " " */
@.nr.nf=. /*undo the above move. */
end /*try different move. */
end /*t*/ /* [↑] all moves tried.*/
return 0 /*tour is not possible. */

output   when using the default input:

A solution for the knight's tour on a  8x8  chessboard:

┌───┬───┬───┬───┬───┬───┬───┬───┐
│-1-│38 │55 │34 │ 3 │36 │19 │22 │
├───┼───┼───┼───┼───┼───┼───┼───┤
│54 │47 │ 2 │37 │20 │23 │ 4 │17 │
├───┼───┼───┼───┼───┼───┼───┼───┤
│39 │56 │33 │46 │35 │18 │21 │10 │
├───┼───┼───┼───┼───┼───┼───┼───┤
│48 │53 │40 │57 │24 │11 │16 │ 5 │
├───┼───┼───┼───┼───┼───┼───┼───┤
│59 │32 │45 │52 │41 │26 │ 9 │12 │
├───┼───┼───┼───┼───┼───┼───┼───┤
│44 │49 │58 │25 │62 │15 │ 6 │27 │
├───┼───┼───┼───┼───┼───┼───┼───┤
│31 │60 │51 │42 │29 │ 8 │13 │end│
├───┼───┼───┼───┼───┼───┼───┼───┤
│50 │43 │30 │61 │14 │63 │28 │ 7 │
└───┴───┴───┴───┴───┴───┴───┴───┘

## Ruby

Knights tour using Warnsdorffs rule

class Board
def self.end=(end_val)
@@end = end_val
end

def try(seq_num)
self.value = seq_num
return true if [email protected]@end
a = []
a << [wdof(cell.adj)*10+n, cell] if cell.value.zero?
end
a.sort.each {|_, cell| return true if cell.try(seq_num+1)}
self.value = 0
false
end

end
end

def initialize(rows, cols)
@rows, @cols = rows, cols
unless defined? ADJACENT # default move (Knight)
end
@board = Array.new(rows+frame) do |i|
Array.new(cols+frame) do |j|
(i<rows and j<cols) ? Cell.new(0) : nil # frame (Sentinel value : nil)
end
end
rows.times do |i|
cols.times do |j|
end
end
Cell.end = rows * cols
@format = " %#{(rows * cols).to_s.size}d"
end

def solve(sx, sy)
if (@rows*@cols).odd? and (sx+sy).odd?
puts "No solution"
else
puts (@board[sx][sy].try(1) ? to_s : "No solution")
end
end

def to_s
(0[email protected]).map do |x|
(0[email protected]).map{|y| @format % @board[x][y].value}.join
end
end
end

def knight_tour(rows=8, cols=rows, sx=rand(rows), sy=rand(cols))
puts "\nBoard (%d x %d), Start:[%d, %d]" % [rows, cols, sx, sy]
Board.new(rows, cols).solve(sx, sy)
end

knight_tour(8,8,3,1)

knight_tour(5,5,2,2)

knight_tour(4,9,0,0)

knight_tour(5,5,0,1)

knight_tour(12,12,1,1)

Which produces:

Board (8 x 8), Start:[3, 1]
23 20  3 32 25 10  5  8
2 35 24 21  4  7 26 11
19 22 33 36 31 28  9  6
34  1 50 29 48 37 12 27
51 18 53 44 61 30 47 38
54 43 56 49 58 45 62 13
17 52 41 60 15 64 39 46
42 55 16 57 40 59 14 63

Board (5 x 5), Start:[2, 2]
19  8  3 14 25
2 13 18  9  4
7 20  1 24 15
12 17 22  5 10
21  6 11 16 23

Board (4 x 9), Start:[0, 0]
1 34  3 28 13 24  9 20 17
4 29  6 33  8 27 18 23 10
35  2 31 14 25 12 21 16 19
30  5 36  7 32 15 26 11 22

Board (5 x 5), Start:[0, 1]
No solution

Board (12 x 12), Start:[1, 1]
87  24  59   2  89  26  61   4  39   8  31   6
58   1  88  25  60   3  92  27  30   5  38   9
23  86  83  90 103  98  29  62  93  40   7  32
82  57 102  99  84  91 104  97  28  37  10  41
101  22  85 114 105 116 111  94  63  96  33  36
56  81 100 123 128 113 106 117 110  35  42  11
21 122 141  80 115 124 127 112  95  64 109  34
144  55  78 121 142 129 118 107 126 133  12  43
51  20 143 140  79 120 125 138  69 108  65 134
54  73  52  77 130 139  70 119 132 137  44  13
19  50  75  72  17  48 131  68  15  46 135  66
74  53  18  49  76  71  16  47 136  67  14  45

## Rust

use std::fmt;

const SIZE: usize = 8;
const MOVES: [(i32, i32); 8] = [
(2, 1),
(1, 2),
(-1, 2),
(-2, 1),
(-2, -1),
(-1, -2),
(1, -2),
(2, -1),
];

#[derive(Copy, Clone, Eq, PartialEq, PartialOrd, Ord)]
struct Point {
x: i32,
y: i32,
}

impl Point {
fn mov(&self, &(dx, dy): &(i32, i32)) -> Self {
Self {
x: self.x + dx,
y: self.y + dy,
}
}
}

struct Board {
field: [[i32; SIZE]; SIZE],
}

impl Board {
fn new() -> Self {
Self {
field: [[0; SIZE]; SIZE],
}
}

fn available(&self, p: Point) -> bool {
0 <= p.x
&& p.x < SIZE as i32
&& 0 <= p.y
&& p.y < SIZE as i32
&& self.field[p.x as usize][p.y as usize] == 0
}

// calculate the number of possible moves
fn count_degree(&self, p: Point) -> i32 {
let mut count = 0;
for dir in MOVES.iter() {
let next = p.mov(dir);
if self.available(next) {
count += 1;
}
}
count
}
}

impl fmt::Display for Board {
fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
for row in self.field.iter() {
for x in row.iter() {
write!(f, "{:3} ", x)?;
}
write!(f, "\n")?;
}
Ok(())
}
}

fn knights_tour(x: i32, y: i32) -> Option<Board> {
let mut board = Board::new();
let mut p = Point { x: x, y: y };
let mut step = 1;
board.field[p.x as usize][p.y as usize] = step;
step += 1;

while step <= (SIZE * SIZE) as i32 {
// choose next square by Warnsdorf's rule
let mut candidates = vec![];
for dir in MOVES.iter() {
}
}
match candidates.iter().min() {
// move to next square
// can't move
None => return None,
};
board.field[p.x as usize][p.y as usize] = step;
step += 1;
}
Some(board)
}

fn main() {
let (x, y) = (3, 1);
println!("Board size: {}", SIZE);
println!("Starting position: ({}, {})", x, y);
match knights_tour(x, y) {
Some(b) => print!("{}", b),
None => println!("Fail!"),
}
}
Output:
Board size: 8
Starting position: (3, 1)
23  20   3  32  25  10   5   8
2  33  24  21   4   7  26  11
19  22  51  34  31  28   9   6
50   1  40  29  54  35  12  27
41  18  55  52  61  30  57  36
46  49  44  39  56  53  62  13
17  42  47  60  15  64  37  58
48  45  16  43  38  59  14  63

## Scala

val b=Seq.tabulate(8,8,8,8)((x,y,z,t)=>(1L<<(x*8+y),1L<<(z*8+t),f"\${97+z}%c\${49+t}%c",(x-z)*(x-z)+(y-t)*(y-t)==5)).flatten.flatten.flatten.filter(_._4).groupBy(_._1)
def f(p:Long,s:Long,v:Any){if(-1L!=s)b(p).foreach(x=>if((s&x._2)==0)f(x._2,s|x._2,v+x._3))else println(v)}
f(1,1,"a1")

a1b3a5b7c5a4b2c4a3b1c3a2b4a6b8c6a7b5c7a8b6c8d6e4d2f1e3c2d4e2c1d3e1g2f4d5e7g8h6f5h4g6h8f7d8e6f8d7e5g4h2f3g1h3g5h7f6e8g7h5g3h1f2d1

## Scheme

;;/usr/bin/petite
;;encoding:utf-8
;;Author:Panda
;;Mail:[email protected]
;;Created Time:Thu 29 Jan 2015 10:18:49 AM CST
;;Description:

;;size of the chessboard
(define X 8)
(define Y 8)
;;position is an integer that could be decoded into the x coordinate and y coordinate
(define(decode position)
(cons (div position Y) (remainder position Y)))
;;record the paths and number of territories you have conquered
(define dictionary '())
(define counter 1)
;;define the forbiddend territories(conquered and cul-de-sac)
(define forbiddened '())
;;renew when havn't conquered the world.
(define (renew position)
(define possible
(let ((rules (list (+ (* 2 Y) 1 position)
(+ (* 2 Y) -1 position)
(+ (* -2 Y) 1 position)
(+ (* -2 Y) -1 position)
(+ Y 2 position)
(+ Y -2 position)
(- position Y 2)
(- position Y -2))))
(filter (lambda(x) (not (or (member x forbiddened) (< x 0) (>= x (* X Y))))) rules)))
(if (null? possible)
(begin (set! forbiddened (cons (car dictionary) forbiddened))
(set! dictionary (cdr dictionary))
(set! counter (- counter 1))
(car dictionary))
(begin (set! dictionary (cons (car possible) dictionary))
(set! forbiddened dictionary)
(set! counter (+ counter 1))
(car possible))))
;;go to search
(define (go position)
(if (= counter (* X Y))
(begin
(set! result (reverse dictionary))
(display (map (lambda(x) (decode x)) result)))
(go (renew position))))

Output:
(go 35)
((6 . 4) (4 . 5) (6 . 6) (4 . 7) (7 . 0) (5 . 1) (7 . 2) (5 . 3) (7 . 4) (5 . 5) (7 . 6) (5 . 7) (4 . 0) (6 . 1) (4 . 2) (6 . 3) (4 . 4) (6 . 5) (4 . 6) (6 . 7) (5 . 0) (7 . 1) (5 . 2) (7 . 3) (5 . 4) (7 . 5) (5 . 6) (7 . 7) (6 . 0) (4 . 1) (6 . 2) (4 . 3) (2 . 4) (0 . 5) (2 . 6) (0 . 7) (3 . 0) (1 . 1) (3 . 2) (1 . 3) (3 . 4) (1 . 5) (3 . 6) (1 . 7) (0 . 0) (2 . 1) (0 . 2) (2 . 3) (0 . 4) (2 . 5) (0 . 6) (2 . 7) (1 . 0) (3 . 1) (1 . 2) (3 . 3) (1 . 4) (3 . 5) (1 . 6) (3 . 7) (2 . 0) (0 . 1) (2 . 2))

## SequenceL

Knights tour using Warnsdorffs rule (No Backtracking)

import <Utilities/Sequence.sl>;
import <Utilities/Conversion.sl>;

main(args(2)) :=
let
N := stringToInt(args[1]) when size(args) > 0 else 8;
M := stringToInt(args[2]) when size(args) > 1 else N;
startX := stringToInt(args[3]) when size(args) > 2 else 1;
startY := stringToInt(args[4]) when size(args) > 3 else 1;
board[i,j] := 0 foreach i within 1 ... N, j within 1 ... M;
spacing := size(toString(N*M)) + 1;
in
join(printRow(
tour(setBoard(board, startX, startX, 1), [startX,startY], 2),
spacing));

potentialMoves := [[2,1], [2,-1], [1,2], [1,-2], [-1,2], [-1,-2], [-2,1], [-2,-1]];

printRow(row(1), spacing) := join(printSquare(row, spacing)) ++ "\n";

printSquare(val, spacing) :=
let
str := toString(val);
in
duplicate(' ', spacing - size(str)) ++ str;

tour(board(2), current(1), move) :=
let
validMoves := validMove(board, current + potentialMoves);
numMoves[i] := size(validMove(board, validMoves[i] + potentialMoves));
chosenMove := minPosition(numMoves);
in
board when move > size(board) * size(board[1]) else
[] when size(validMoves) = 0 else
[] when move < size(board) * size(board[1]) and numMoves[chosenMove] = 0 else
tour(setBoard(board, validMoves[chosenMove][1], validMoves[chosenMove][2], move), validMoves[chosenMove], move + 1);

validMove(board(2), position(1)) :=
(position when board[position[1], position[2]] = 0)
when position[1] >= 1 and position[1] <= size(board) and position[2] >= 1 and position[2] <= size(board);

minPosition(x(1)) := minPositionHelper(x, 2, 1, x[1]);
minPositionHelper(x(1), i, minPos, minVal) :=
minPos when i > size(x) else
minPositionHelper(x, i + 1, minPos, minVal) when x[i] > minVal else
minPositionHelper(x, i + 1, i, x[i]);

setBoard(board(2), x, y, value)[i,j] :=
value when x = i and y = j else
board[i,j] foreach i within 1 ... size(board), j within 1 ... size(board[1]);

Output:

8 X 8 board:

1 16 31 40  3 18 21 56
30 39  2 17 42 55  4 19
15 32 41 46 53 20 57 22
38 29 48 43 58 45 54  5
33 14 37 52 47 60 23 62
28 49 34 59 44 63  6  9
13 36 51 26 11  8 61 24
50 27 12 35 64 25 10  7

20 X 20 board:

1  40  81  90   3  42  77  94   5  44  73 102   7  46  69  62   9  48  51  60
82  89   2  41  92  95   4  43  76 101   6  45  72 103   8  47  68  61  10  49
39  80  91  96 153  78  93 100 129  74 109 104 123  70 111 120  63  50  59  52
88  83 154  79  98 159 152  75 108 105 128  71 110 121 124  67 112 119  64  11
155  38  97 160 157 200  99 162 151 130 107 122 127 132 141 118 125  66  53  58
84  87 156 199 176 161 158 201 106 163 150 131 142 145 126 133 140 113  12  65
37 182  85 178 207 198 175 164 173 216 143 166 149 222 139 146 117 134  57  54
86 179 206 197 204 177 208 217 202 165 172 221 144 167 148 223 138  55 114  13
183  36 181 212 209 218 203 174 215 220 227 170 281 224 303 168 147 116 135  56
180 211 196 205 230 213 238 219 228 171 280 225 302 169 282 343 304 137  14 115
35 184 231 210 237 246 229 214 279 226 301 298 283 342 367 308 347 344 305 136
232 195 236 245 234 239 278 247 300 297 284 359 366 309 348 345 368 307 350  15
185  34 233 240 261 248 287 296 285 358 299 310 341 378 365 384 349 346 369 306
194 241 250 235 244 277 260 313 294 311 360 373 364 383 354 379 370 385  16 351
33 186 243 262 249 288 295 286 361 316 357 340 377 372 395 386 353 380 333 388
242 193 254 251 276 259 314 293 312 321 374 363 398 355 382 371 394 387 352  17
187  32 263 258 267 252 289 322 315 362 317 356 339 376 399 396 381 334 389 332
192 255 190 253 264 275 268 271 292 323 320 375 326 397 338 335 390 393  18  21
31 188 257 266  29 270 273 290  27 318 327 324  25 336 329 400  23  20 331 392
256 191  30 189 274 265  28 269 272 291  26 319 328 325  24 337 330 391  22  19

## Sidef

Translation of: Raku
var board = []
var I = 8
var J = 8
var F = (I*J > 99 ? '%3d' : '%2d')

var (i, j) = (I.irand, J.irand)

func from_algebraic(square) {
if (var match = square.match(/^([a-z])([0-9])\z/)) {
return(I - Num(match[1]), match[0].ord - 'a'.ord)
}
die "Invalid block square: #{square}"
}

func possible_moves(i,j) {
gather {
for ni,nj in [
[i-2,j-1], [i-2,j+1], [i-1,j-2], [i-1,j+2],
[i+1,j-2], [i+1,j+2], [i+2,j-1], [i+2,j+1],
] {
if ((ni ~~ ^I) && (nj ~~ ^J) && !board[ni][nj]) {
take([ni, nj])
}
}
}
}

func to_algebraic(i,j) {
('a'.ord + j).chr + Str(I - i)
}

if (ARGV[0]) {
(i, j) = from_algebraic(ARGV[0])
}

var moves = []
for move in (1 .. I*J) {
moves << to_algebraic(i, j)
board[i][j] = move
var min = [9]
for target in possible_moves(i, j) {
var (ni, nj) = target...
var nxt = possible_moves(ni, nj).len
if (nxt < min[0]) {
min = [nxt, ni, nj]
}
}

(i, j) = min[1,2]
}

say (moves/4 -> map { .join(', ') }.join("\n") + "\n")

for i in ^I {
for j in ^J {
(i%2 == j%2) && print "\e[7m"
F.printf(board[i][j])
print "\e[0m"
}
print "\n"
}

## Swift

Translation of: Rust
public struct CPoint {
public var x: Int
public var y: Int

public init(x: Int, y: Int) {
(self.x, self.y) = (x, y)
}

public func move(by: (dx: Int, dy: Int)) -> CPoint {
return CPoint(x: self.x + by.dx, y: self.y + by.dy)
}
}

extension CPoint: Comparable {
public static func <(lhs: CPoint, rhs: CPoint) -> Bool {
if lhs.x == rhs.x {
return lhs.y < rhs.y
} else {
return lhs.x < rhs.x
}
}
}

public class KnightsTour {
public var size: Int { board.count }

private var board: [[Int]]

public init(size: Int) {
board = Array(repeating: Array(repeating: 0, count: size), count: size)
}

public func countMoves(forPoint point: CPoint) -> Int {
return KnightsTour.knightMoves.lazy
.map(point.move)
.reduce(0, {count, movedTo in
return squareAvailable(movedTo) ? count + 1 : count
})
}

public func printBoard() {
for row in board {
for x in row {
print("\(x) ", terminator: "")
}

print()
}

print()
}

private func reset() {
for i in 0..<size {
for j in 0..<size {
board[i][j] = 0
}
}
}

public func squareAvailable(_ p: CPoint) -> Bool {
return 0 <= p.x
&& p.x < size
&& 0 <= p.y
&& p.y < size
&& board[p.x][p.y] == 0
}

public func tour(startingAt point: CPoint = CPoint(x: 0, y: 0)) -> Bool {
var step = 2
var p = point

reset()

board[p.x][p.y] = 1

while step <= size * size {
let candidates = KnightsTour.knightMoves.lazy
.map(p.move)
.map({moved in (moved, self.countMoves(forPoint: moved), self.squareAvailable(moved)) })
.filter({ \$0.2 })

guard let bestMove = candidates.sorted(by: bestChoice).first else {
return false
}

p = bestMove.0
board[p.x][p.y] = step

step += 1
}

return true
}
}

private func bestChoice(_ choice1: (CPoint, Int, Bool), _ choice2: (CPoint, Int, Bool)) -> Bool {
if choice1.1 == choice2.1 {
return choice1.0 < choice2.0
}

return choice1.1 < choice2.1
}

extension KnightsTour {
fileprivate static let knightMoves = [
(2, 1),
(1, 2),
(-1, 2),
(-2, 1),
(-2, -1),
(-1, -2),
(1, -2),
(2, -1),
]
}

let b = KnightsTour(size: 8)

print()

let completed = b.tour(startingAt: CPoint(x: 3, y: 1))

if completed {
print("Completed tour")
} else {
print("Did not complete tour")
}

b.printBoard()
Output:
Completed tour
23 20 3 32 25 10 5 8
2 33 24 21 4 7 26 11
19 22 51 34 31 28 9 6
50 1 40 29 54 35 12 27
41 18 55 52 61 30 57 36
46 49 44 39 56 53 62 13
17 42 47 60 15 64 37 58
48 45 16 43 38 59 14 63

## Tcl

package require Tcl 8.6;    # For object support, which makes coding simpler

oo::class create KnightsTour {
variable width height visited

constructor {{w 8} {h 8}} {
set width \$w
set height \$h
set visited {}
}

method ValidMoves {square} {
lassign \$square c r
set moves {}
foreach {dx dy} {-1 -2 -2 -1 -2 1 -1 2 1 2 2 1 2 -1 1 -2} {
set col [expr {(\$c % \$width) + \$dx}]
set row [expr {(\$r % \$height) + \$dy}]
if {\$row >= 0 && \$row < \$height && \$col >=0 && \$col < \$width} {
lappend moves [list \$col \$row]
}
}
return \$moves
}

method CheckSquare {square} {
set moves 0
foreach site [my ValidMoves \$square] {
if {\$site ni \$visited} {
incr moves
}
}
return \$moves
}

method Next {square} {
set minimum 9
set nextSquare {-1 -1}
foreach site [my ValidMoves \$square] {
if {\$site ni \$visited} {
set count [my CheckSquare \$site]
if {\$count < \$minimum} {
set minimum \$count
set nextSquare \$site
} elseif {\$count == \$minimum} {
set nextSquare [my Edgemost \$nextSquare \$site]
}
}
}
return \$nextSquare
}

method Edgemost {a b} {
lassign \$a ca ra
lassign \$b cb rb
# Calculate distances to edge
set da [expr {min(\$ca, \$width - 1 - \$ca, \$ra, \$height - 1 - \$ra)}]
set db [expr {min(\$cb, \$width - 1 - \$cb, \$rb, \$height - 1 - \$rb)}]
if {\$da < \$db} {return \$a} else {return \$b}
}

method FormatSquare {square} {
lassign \$square c r
format %c%d [expr {97 + \$c}] [expr {1 + \$r}]
}

method constructFrom {initial} {
while 1 {
set visited [list \$initial]
set square \$initial
while 1 {
set square [my Next \$square]
if {\$square eq {-1 -1}} {
break
}
lappend visited \$square
}
if {[llength \$visited] == \$height*\$width} {
return
}
puts stderr "rejecting path of length [llength \$visited]..."
}
}

method constructRandom {} {
my constructFrom [list \
[expr {int(rand()*\$width)}] [expr {int(rand()*\$height)}]]
}

method print {} {
set s " "
foreach square \$visited {
puts -nonewline "\$s[my FormatSquare \$square]"
if {[incr i]%12} {
set s " -> "
} else {
set s "\n -> "
}
}
puts ""
}

method isClosed {} {
set a [lindex \$visited 0]
set b [lindex \$visited end]
expr {\$a in [my ValidMoves \$b]}
}
}

Demonstrating:

set kt [KnightsTour new]
\$kt constructRandom
\$kt print
if {[\$kt isClosed]} {
puts "This is a closed tour"
} else {
puts "This is an open tour"
}

Sample output:

e6 -> f8 -> h7 -> g5 -> h3 -> g1 -> e2 -> c1 -> a2 -> b4 -> a6 -> b8
-> d7 -> b6 -> a8 -> c7 -> e8 -> g7 -> h5 -> g3 -> h1 -> f2 -> d1 -> b2
-> a4 -> c3 -> b1 -> a3 -> b5 -> a7 -> c8 -> e7 -> g8 -> h6 -> f7 -> h8
-> g6 -> h4 -> g2 -> f4 -> d5 -> f6 -> g4 -> h2 -> f1 -> e3 -> f5 -> d6
-> e4 -> d2 -> c4 -> a5 -> b7 -> d8 -> c6 -> e5 -> f3 -> e1 -> d3 -> c5
-> b3 -> a1 -> c2 -> d4
This is a closed tour

The above code supports other sizes of boards and starting from nominated locations:

set kt [KnightsTour new 7 7]
\$kt constructFrom {0 0}
\$kt print
if {[\$kt isClosed]} {
puts "This is a closed tour"
} else {
puts "This is an open tour"
}

Which could produce this output:

a1 -> c2 -> e1 -> g2 -> f4 -> g6 -> e7 -> f5 -> g7 -> e6 -> g5 -> f7
-> d6 -> b7 -> a5 -> b3 -> c1 -> a2 -> b4 -> a6 -> c7 -> b5 -> a7 -> c6
-> d4 -> e2 -> g1 -> f3 -> d2 -> f1 -> g3 -> e4 -> f2 -> g4 -> f6 -> d7
-> e5 -> d3 -> c5 -> a4 -> b2 -> d1 -> e3 -> d5 -> b6 -> c4 -> a3 -> b1
-> c3
This is an open tour

## Wren

Translation of: Kotlin
class Square {
construct new(x, y) {
_x = x
_y = y
}

x { _x }
y { _y }

==(other) { _x == other.x && _y == other.y }
}

var board = List.filled(8 * 8, null)
for (i in 0...board.count) board[i] = Square.new((i/8).floor + 1, i%8 + 1)
var axisMoves = [1, 2, -1, -2]

var allPairs = Fn.new { |a|
var pairs = []
for (i in a) {
for (j in a) pairs.add([i, j])
}
return pairs
}

var knightMoves = Fn.new { |s|
var moves = allPairs.call(axisMoves).where { |p| p[0].abs != p[1].abs }
var onBoard = Fn.new { |s| board.any { |i| i == s } }
return moves.map { |p| Square.new(s.x + p[0], s.y + p[1]) }.where(onBoard)
}

var knightTour // recursive
knightTour = Fn.new { |moves|
var findMoves = Fn.new { |s|
return knightMoves.call(s).where { |m| !moves.any { |m2| m2 == m } }.toList
}
var fm = findMoves.call(moves[-1])
if (fm.isEmpty) return moves
var lowest = findMoves.call(fm[0]).count
var lowestIndex = 0
for (i in 1...fm.count) {
var count = findMoves.call(fm[i]).count
if (count < lowest) {
lowest = count
lowestIndex = i
}
}
var newSquare = fm[lowestIndex]
return knightTour.call(moves + [newSquare])
}

var knightTourFrom = Fn.new { |start| knightTour.call([start]) }

var col = 0
for (p in knightTourFrom.call(Square.new(1, 1))) {
System.write("%(p.x),%(p.y)")
System.write((col == 7) ? "\n" : " ")
col = (col + 1) % 8
}
Output:
1,1  2,3  3,1  1,2  2,4  1,6  2,8  4,7
6,8  8,7  7,5  8,3  7,1  5,2  7,3  8,1
6,2  4,1  2,2  1,4  2,6  1,8  3,7  5,8
7,7  8,5  6,6  7,8  8,6  7,4  8,2  6,1
4,2  2,1  3,3  5,4  3,5  4,3  5,1  6,3
8,4  7,2  6,4  5,6  4,8  2,7  1,5  3,6
1,7  3,8  5,7  4,5  5,3  6,5  4,4  3,2
1,3  2,5  4,6  3,4  5,5  6,7  8,8  7,6

## XPL0

int     Board(8+2+2, 8+2+2);            \board array with borders
int LegalX, LegalY; \arrays of legal moves
def IntSize=4; \number of bytes in an integer (4 or 2)
include c:\cxpl\codes; \intrinsic 'code' declarations

func Try(I, X, Y); \Make a tentative move from X,Y
int I, X, Y;
int K, U, V;
[for K:= 0 to 8-1 do \for all possible moves...
[U:= X + LegalX(K); \U and V are next square
V:= Y + LegalY(K);
if Board(U,V) = 0 then \if square has not been visited then
[Board(U,V):= I; \ mark square with sequence number
if I = 8*8 then return true;
if Try(I+1, U, V) then return true \led to solution?
else Board(U,V):= 0; \no, undo tenative move
];
];
return false;
]; \Try

int I, J;
[LegalX:= [2, 1, -1, -2, -2, -1, 1, 2];
LegalY:= [1, 2, 2, 1, -1, -2, -2, -1];

for J:= 0 to 8+2+2-1 do \set up surrounding border for speed
for I:= 0 to 8+2+2-1 do
Board(I,J):= 1;
for J:= 0 to 8+2+2-1 do \reposition Board(0,0) to Board(2,2)
Board(J):= Board(J) + 2*IntSize;
Board:= Board + 2*IntSize;
for J:= 0 to 8-1 do \empty board
for I:= 0 to 8-1 do
Board(I,J):= 0;
Text(0, "Starting square (1-8,1-8): "); I:= IntIn(0)-1; J:= IntIn(0)-1;
Board(I,J):= 1; \starting location is 0,0

if Try(2, I, J) then \try to find second square
[for J:= 0 to 8-1 do \draw board with knight's move sequence
[for I:= 0 to 8-1 do
[if Board(I,J) < 10 then ChOut(0, ^ );
IntOut(0, Board(I,J));
ChOut(0, ^ );
];
CrLf(0);
];
]
else Text(0, "No Solution.^M^J");
]

Example output:

Starting square (1-8,1-8): 1 1
1 38 59 36 43 48 57 52
60 35  2 49 58 51 44 47
39 32 37 42  3 46 53 56
34 61 40 27 50 55  4 45
31 10 33 62 41 26 23 54
18 63 28 11 24 21 14  5
9 30 19 16  7 12 25 22
64 17  8 29 20 15  6 13

## XSLT

This solution is for XSLT 3.0 Working Draft 10 (July 2012). This solution, originally reported on this blog post, will be updated or removed when the final version of XSLT 3.0 is released.

First we build a generic package for solving any kind of tour over the chess board. Here it is…

<xsl:package xsl:version="3.0"
xmlns:xsl="http://www.w3.org/1999/XSL/Transform"
xmlns:xs="http://www.w3.org/2001/XMLSchema"
xmlns:fn="http://www.w3.org/2005/xpath-functions"
xmlns:tour="http://www.seanbdurkin.id.au/tour"
name="tour:tours">
<xsl:stylesheet>
<xsl:function name="tour:manufacture-square"
as="element(square)" visibility="public">
<xsl:param name="rank" as="xs:integer" />
<xsl:param name="file" as="xs:integer" />
<square file="\$file" rank="\$rank" />
</xsl:function>

<xsl:function name="tour:on-board" as="xs:boolean" visibility="public">
<xsl:param name="rank" as="xs:integer" />
<xsl:param name="file" as="xs:integer" />
<xsl:copy-of select="(\$rank ge 1) and (\$rank le 8) and
(\$file ge 1) and (\$file le 8)" />
</xsl:function>

<xsl:function name="tour:solve-tour" as="item()*" visibility="public">
<!-- Solves the tour for any specified piece. -->
<!-- Outputs either a full solution of 64 squares, of if fail,
a copy of the \$state input. -->
<xsl:param name="state" as="item()+" />
<xsl:variable name="compute-possible-moves"
select="\$state[. instance of function(*)]"
as="function(element(square)) as element(square)*">
<xsl:variable name="way-points" select="\$state/self::square" />
<xsl:choose>
<xsl:when test="count(\$way-points) eq 64">
<xsl:sequence ="\$state" />
</xsl:when>
<xsl:otherwise>
<xsl:sequence select="
let \$try-move := function( \$state as item()*, \$move as item()) as item()*)
{
if \$state/self::square[@file=\$move/@file]
[@rank=\$move/@rank]
then \$state
else tour:solve-tour( ( \$state, \$move) )
},
\$possible-moves := \$compute-possible-moves( \$way-points[last()])
return if empty( \$possible-moves) then \$state
else fn:fold-left( \$try-move, \$state, \$possible-moves)" />
</xsl:otherwise>
</xsl:choose>
</xsl:variable></xsl:function>
</xsl:stylesheet>

<xsl:expose component="function"
names="tour:manufacture-square tour:on-board tour:solve-tour"
visibility="public" />

</xsl:package>

And now for the style-sheet to solve the Knight’s tour…

<xsl:stylesheet version="3.0"
xmlns:xsl="http://www.w3.org/1999/XSL/Transform"
xmlns:xs="http://www.w3.org/2001/XMLSchema"
xmlns:fn="http://www.w3.org/2005/xpath-functions"
xmlns:tour="http://www.seanbdurkin.id.au/tour"
exclude-result-prefixes="xsl fn xs tour">
<xsl:use-package name="tour:tours" />
<xsl:output indent="yes" encoding="UTF-8" omit-xml-declaration="yes" />
<xsl:mode on-no-match="shallow-copy" streamable="yes"/>

<xsl:template match="knight[square]">
<xsl:variable name="error">
<error>Failed to find solution to Knight's Tour.</error>
</xsl:variable>
<xsl:copy>
<xsl:copy-of select="
let \$final-state := tour:solve-tour((
function( \$piece-position as element(square)) as element(square)*
{ (: This function defines a knight's move. :)
let \$r0 := number( \$piece-position/@rank),
let \$f0 := number( \$piece-position/@file),
for \$r in -2..2, \$f in -2..2 return
if (abs(\$r) + abs(\$f) eq 3) and
tour:on-board(\$r+\$r0, \$f+\$f0) then
tour:manufacture-square(\$r+\$r0, \$f+\$f0)
else ()
}
, current()/square)),
\$solution := \$final-state/self::square
return if count(\$solution) eq 64 then \$solution
else \$error/*" />
</xsl:copy>
</xsl:template>

<!-- Add templates for other piece types if you want to solve
their tours too. Solve by calling tour:solve-tour() . -->

</xsl:stylesheet>

So an input like this…

<tt>
<knight>
<square file="1" rank="1" />
</knight>
</tt>

…should be transformed in something like this…

<tt>
<knight>
<square file="1" rank="1" />
<square file="2" rank="3" />
<square file="1" rank="5" />
... etc for 64 squares.
</knight>
</tt>

## zkl

// Use Warnsdorff's rule to perform a knights tour of a 8x8 board in
// linear time.
// See Pohl, Ira (July 1967),
// "A method for finding Hamilton paths and Knight's tours"
// http://portal.acm.org/citation.cfm?id=363463
// Uses back tracking as a tie breaker (for the few cases in a 8x8 tour)
class Board{
var[const]deltas=[[(dx,dy); T(-2,2); T(-1,1); _]].extend(
[[(dx,dy); T(-1,1); T(-2,2); _]]);
fcn init{
var board=L();
(0).pump(64,board.append.fpM("1-",Void)); // fill board with Void
}
fcn idx(x,y) { x*8+y }
fcn isMoveOK(x,y){ (0<=x<8) and (0<=y<8) and Void==board[idx(x,y)] }
fcn gyrate(x,y,f){ // walk all legal moves from (a,b)
deltas.pump(List,'wrap([(dx,dy)]){
x+=dx; y+=dy; if(isMoveOK(x,y)) f(x,y); else Void.Skip
});
}
fcn count(x,y){ n:=Ref(0); gyrate(x,y,n.inc); n.value }
fcn moves(x,y){ gyrate(x,y,fcn(x,y){ T(x,y,count(x,y)) })}
fcn knightsTour(x=0,y=0,n=1){ // using Warnsdorff's rule
board[idx(x,y)]=n;
while(m:=moves(x,y)){
min:=m.reduce('wrap(pc,[(_,_,c)]){ (pc<c) and pc or c },9);
m=m.filter('wrap([(_,_,c)]){ c==min }); // moves with same min moves
if(m.len()>1){ // tie breaker time, may need to backtrack
bs:=board.copy();
if (64==m.pump(Void,'wrap([(a,b)]){
board[idx(a,b)]=n;
n2:=knightsTour(a,b,n+1);
if (n2==64) return(Void.Stop,n2); // found a solution
board=bs.copy();
})) return(64);
return(0);
}
else{
x,y=m[0]; n+=1;
board[idx(x,y)]=n;
}
} //while
return(n);
}
fcn(ns){ vm.arglist.apply("%2s".fmt).concat(",")+"\n" });
}
}
b:=Board(); b.knightsTour(3,3);
b.println();
Output:
3,34, 5,54,19,36,29,50
6,21, 2,35,56,49,18,37
33, 4,55,20,53,30,51,28
22, 7,32, 1,48,57,38,17
11,44,23,62,31,52,27,58
8,63,10,45,60,47,16,39
43,12,61,24,41,14,59,26
64, 9,42,13,46,25,40,15

Check that a solution for all squares is found:

[[(x,y); [0..7]; [0..7];
{ b:=Board(); n:=b.knightsTour(x,y); if(n!=64) b.println(">>>",x,",",y) } ]];
Output: